5Solving Systems of Linear Equations - Big Ideas …...5Solving Systems of Linear Equations 5.1 Solving Systems of Linear Equations by Graphing 5.2 Solving Systems of Linear Equations

5 Solving Systems of Linear Equations

5.1 Solving Systems of Linear Equations by Graphing

5.2 Solving Systems of Linear Equations by Substitution

5.3 Solving Systems of Linear Equations by Elimination

5.4 Solving Special Systems of Linear Equations

5.5 Solving Equations by Graphing

5.6 Linear Inequalities in Two Variables

5.7 Systems of Linear Inequalities

Fishing (p. 265)

Pets (p. 250)

Drama Club (p. 228)

Fruit Salad (p. 255)

Roofing Contractor (p. 222)

Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

Drama Club (p 228)

RRo ffofiing CCo tntra tctor (((p. 2222 )2)2)

y q

FiFi hshiing ((p. 26265)5)

SEE the Big Idea

FFr iuitt SSalladd ((p 25255)5)

217

Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyGraphing Linear Functions (A.3.C)

Example 1 Graph 3 + y = 1 — 2 x.

Step 1 Rewrite the equation in slope-intercept form.

y = 1 —

2 x − 3

Step 2 Find the slope and the y-intercept.

m = 1 —

2 and b = −3

Step 3 The y-intercept is −3. So, plot (0, −3).

Step 4 Use the slope to find another point on the line.

slope = rise

— run

= 1 —

2

Plot the point that is 2 units right and 1 unit up from (0, −3). Draw a line

through the two points.

Graph the equation.

1. y + 4 = x 2. 6x − y = −1 3. 4x + 5y = 20 4. −2y + 12 = −3x

Solving and Graphing Linear Inequalities (7.10.B, A.5.B)

Example 2 Solve 2x − 17 ≤ 8x − 5. Graph the solution.

2x − 17 ≤ 8x − 5 Write the inequality.

+ 5 + 5 Add 5 to each side.

2x − 12 ≤ 8x Simplify.

− 2x − 2x Subtract 2x from each side.

−12 ≤ 6x Simplify.

−12

— 6 ≤ 6x

— 6 Divide each side by 6.

−2 ≤ x Simplify.

The solution is x ≥ −2.

0−5 −4 −3 −2 −1 1 32

x ≥ –2

Solve the inequality. Graph the solution.

5. m + 4 > 9 6. 24 ≤ −6t 7. 2a − 5 ≤ 13

8. −5z + 1 < −14 9. 4k − 16 < k + 2 10. 7w + 12 ≥ 2w − 3

11. ABSTRACT REASONING The graphs of the linear functions g and h have different slopes. The

value of both functions at x = a is b. When g and h are graphed in the same coordinate plane,

what happens at the point (a, b)?

x

y2

−1

−4

42−2−4

1(0, −3)2

218 Chapter 5 Solving Systems of Linear Equations

Using a Graphing Calculator

Mathematical Mathematical ThinkingThinking

Monitoring ProgressMonitoring ProgressUse a graphing calculator to fi nd the point of intersection of the graphs of the two linear equations.

1. y = −2x − 3 2. y = −x + 1 3. 3x − 2y = 2

y = 1 —

2 x − 3 y = x − 2 2x − y = 2

Mathematically profi cient students select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. (A.1.C)

Core Core ConceptConceptFinding the Point of Intersection You can use a graphing calculator to fi nd the point of intersection, if it exists, of

the graphs of two linear equations.

1. Enter the equations into a graphing calculator.

2. Graph the equations in an appropriate viewing window, so that the point of

intersection is visible.

3. Use the intersect feature of the graphing calculator to fi nd the point of

intersection.


Use a graphing calculator to fi nd the point of intersection, if it exists, of the graphs of

the two linear equations.

y = − 1 — 2 x + 2 Equation 1

y = 3x − 5 Equation 2

SOLUTION

The slopes of the lines are not the same, so

you know that the lines intersect. Enter the

equations into a graphing calculator. Then

graph the equations in an appropriate

viewing window.

Use the intersect feature to fi nd the point

of intersection of the lines.

The point of intersection is (2, 1).

−6

−4

4

66

y = − x + 212

y = 3x − 5

−6

−4

4

6

IntersectionX=2 Y=1

Section 5.1 Solving Systems of Linear Equations by Graphing 219

5.1 Solving Systems of Linear Equations by Graphing

Writing a System of Linear Equations

Work with a partner. Your family opens a bed-and-breakfast. They spend $600

preparing a bedroom to rent. The cost to your family for food and utilities is

$15 per night. They charge $75 per night to rent the bedroom.

a. Write an equation that represents the costs.

Cost, C

(in dollars) =

$15 per

night ⋅ Number of

nights, x + $600

b. Write an equation that represents the revenue (income).

Revenue, R

(in dollars) =

$75 per

night ⋅ Number of

nights, x

c. A set of two (or more) linear equations is called a system of linear equations.

Write the system of linear equations for this problem.

Essential QuestionEssential Question How can you solve a system of linear

equations?

Using a Table or Graph to Solve a System

Work with a partner. Use the cost and revenue equations from Exploration 1 to

determine how many nights your family needs to rent the bedroom before recovering

the cost of preparing the bedroom. This is the break-even point.

a. Copy and complete the table.

b. How many nights does your family need to rent the bedroom before breaking even?

c. In the same coordinate plane, graph the cost equation and the revenue equation

from Exploration 1.

d. Find the point of intersection of the two graphs. What does this point represent?

How does this compare to the break-even point in part (b)? Explain.

Communicate Your AnswerCommunicate Your Answer 3. How can you solve a system of linear equations? How can you check your

solution?

4. Solve each system by using a table or sketching a graph. Explain why you chose

each method. Use a graphing calculator to check each solution.

a. y = −4.3x − 1.3 b. y = x c. y = −x − 1

y = 1.7x + 4.7 y = −3x + 8 y = 3x + 5

x (nights) 0 1 2 3 4 5 6 7 8 9 10 11

C (dollars)

R (dollars)

APPLYING MATHEMATICS

To be profi cient in math, you need to identify important quantities in real-life problems and map their relationships using tools such as diagrams, tables, and graphs.

A.2.IA.3.FA.3.GA.5.C

TEXAS ESSENTIAL KNOWLEDGE AND SKILLS


5.1 Lesson What You Will LearnWhat You Will Learn Check solutions of systems of linear equations.

Solve systems of linear equations by graphing.

Use systems of linear equations to solve real-life problems.

Systems of Linear Equationssystem of linear equations, p. 220solution of a system of linear equations, p. 220

Previouslinear equationordered pair

Core VocabularyCore Vocabullarry

Checking Solutions

Tell whether the ordered pair is a solution of the system of linear equations.

a. (2, 5); x + y = 7 Equation 12x − 3y = −11 Equation 2

b. (−2, 0); y = −2x − 4 Equation 1y = x + 4 Equation 2

SOLUTION

a. Substitute 2 for x and 5 for y in each equation.

Equation 1 Equation 2

x + y = 7 2x − 3y = −11

2 + 5 =?

7 2(2) − 3(5) =?

−11

7 = 7 ✓ −11 = −11 ✓ Because the ordered pair (2, 5) is a solution of each equation, it is a solution of

the linear system.

b. Substitute −2 for x and 0 for y in each equation.


y = −2x − 4 y = x + 4

0 =?

−2(−2) − 4 0 =?

−2 + 4

0 = 0 ✓ 0 ≠ 2 ✗ The ordered pair (−2, 0) is a solution of the fi rst equation, but it is not a solution

of the second equation. So, (−2, 0) is not a solution of the linear system.

Monitoring ProgressMonitoring Progress Help in English and Spanish at BigIdeasMath.com

Tell whether the ordered pair is a solution of the system of linear equations.

1. (1, −2); 2x + y = 0

−x + 2y = 5 2. (1, 4);

y = 3x + 1

y = −x + 5

READINGA system of linear equations is also called a linear system.

A system of linear equations is a set of two or more linear equations in the same

variables. An example is shown below.

x + y = 7 Equation 1

2x − 3y = −11 Equation 2

A solution of a system of linear equations in two variables is an ordered pair that is a

solution of each equation in the system.


Solving Systems of Linear Equations by GraphingThe solution of a system of linear equations is the point of intersection of the graphs of

the equations.

Core Core ConceptConceptSolving a System of Linear Equations by GraphingStep 1 Graph each equation in the same coordinate plane.

Step 2 Estimate the point of intersection.

Step 3 Check the point from Step 2 by substituting for x and y in each equation

of the original system.

Solving a System of Linear Equations by Graphing

Solve the system of linear equations by graphing.

y = −2x + 5 Equation 1


SOLUTION

Step 1 Graph each equation.


The graphs appear to intersect at (1, 3).

Step 3 Check your point from Step 2.


y = −2x + 5 y = 4x − 1

3 =?

−2(1) + 5 3 =?

4(1) − 1

3 = 3 ✓ 3 = 3 ✓ The solution is (1, 3).

Check



3. y = x − 2 4. y = 1 —

2 x + 3 5. 2x + y = 5

y = −x + 4 y = − 3 — 2 x − 5 3x − 2y = 4

REMEMBERNote that the linear equations are in slope-intercept form. You can use the method presented in Section 3.5 to graph the equations.

x

y

2

42−2−4

(1, 3)y = −2x + 5

1 3)

y = 4x − 1

−1

−6

−2

6

6

y = 4x − 1y = −2x + 5

IntersectionX=1 Y=3


Modeling with Mathematics

A roofi ng contractor buys 30 bundles of shingles and 4 rolls of roofi ng paper for

$1040. In a second purchase (at the same prices), the contractor buys 8 bundles of

shingles for $256. Find the price per bundle of shingles and the price per roll of

roofi ng paper.

SOLUTION

1. Understand the Problem You know the total price of each purchase and how

many of each item were purchased. You are asked to fi nd the price of each item.

2. Make a Plan Use a verbal model to write a system of linear equations that

represents the problem. Then solve the system of linear equations.

3. Solve the Problem

Words 30 ⋅ Price per

bundle + 4 ⋅

Price

per roll = 1040

8 ⋅ Price per

bundle + 0 ⋅

Price

per roll = 256

Variables Let x be the price (in dollars) per bundle and let y be the

price (in dollars) per roll.

System 30x + 4y = 1040 Equation 1

8x = 256 Equation 2

Step 1 Graph each equation. Note that only

the fi rst quadrant is shown because

x and y must be positive.

Step 2 Estimate the point of intersection. The

graphs appear to intersect at (32, 20).



30x + 4y = 1040 8x = 256

30(32) + 4(20) =?

1040 8(32) =?

256

1040 = 1040 ✓ 256 = 256 ✓ The solution is (32, 20). So, the price per bundle of shingles is $32, and the

price per roll of roofi ng paper is $20.

4. Look Back You can use estimation to check that your solution is reasonable.

A bundle of shingles costs about $30. So, 30 bundles of shingles and 4 rolls of

roofi ng paper (at $20 per roll) cost about 30(30) + 4(20) = $980, and 8 bundles

of shingles costs about 8(30) = $240. These prices are close to the given values,

so the solution seems reasonable.


6. You have a total of 18 math and science exercises for homework. You have

six more math exercises than science exercises. How many exercises do you

have in each subject?

Solving Real-Life Problems

A

$

s

r

S

1

2

3

8 16 240 32 x

80

160

240

320

0

y

(32, 20)

y = −7.5x + 260

x = 32


Tutorial Help in English and Spanish at BigIdeasMath.comExercises5.1

Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3–8, tell whether the ordered pair is a solution of the system of linear equations. (See Example 1.)

3. (2, 6); x + y = 8

3x − y = 0 4. (8, 2);

x − y = 6

2x − 10y = 4

5. (−1, 3); y = −7x − 4

y = 8x + 5

6. (−4, −2); y = 2x + 6

y = −3x − 14

7. (−2, 1); 6x + 5y = −7

2x − 4y = −8 8. (5, −6);

6x + 3y = 12

4x + y = 14

In Exercises 9–12, use the graph to solve the system of linear equations. Check your solution.

9. x − y = 4 10. x + y = 5

4x + y = 1 y − 2x = −4

x

y

−2

42

x

y

2

4

41

11. 6y + 3x = 18 12. 2x − y = −2

−x + 4y = 24 2x + 4y = 8

x

y

2

4

−2−4−6

x

y

4

2−2

In Exercises 13–20, solve the system of linear equations by graphing. (See Example 2.)

13. y = −x + 7 14. y = −x + 4

y = x + 1 y = 2x − 8

15. y = 1 —

3 x + 2 16. y =

3 —

4 x − 4

y = 2 —

3 x + 5 y = −

1 — 2 x + 11

17. 9x + 3y = −3 18. 4x − 4y = 20

2x − y = −4 y = −5

19. x − 4y = −4 20. 3y + 4x = 3

−3x − 4y = 12 x + 3y = −6

ERROR ANALYSIS In Exercises 21 and 22, describe and correct the error in solving the system of linear equations.

21. The solution of

the linear system

x − 3y = 6 and

2x − 3y = 3

is (3, −1).

✗x

y

−1

2

2

22. The solution of

the linear system

y = 2x − 1 and

y = x + 1

is x = 2.

✗

x

y

2

4

42

1. VOCABULARY Do the equations 5y − 2x = 18 and 6x = −4y − 10 form a system of linear

equations? Explain.

2. DIFFERENT WORDS, SAME QUESTION Consider the system of linear equations −4x + 2y = 4

and 4x − y = −6. Which is different? Find “both” answers.

Solve the system of linear equations. Solve each equation for y.

Find the point of intersection

of the graphs of the equations.

Find an ordered pair that is a solution

of each equation in the system.

Vocabulary and Core Concept CheckVocabulary and Core Concept Check


Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the literal equation for y. (Section 1.4)

34. 10x + 5y = 5x + 20 35. 9x + 18 = 6y − 3x 36. 3 — 4 x +

1 —

4 y = 5

Reviewing what you learned in previous grades and lessons

USING TOOLS In Exercises 23–26, use a graphing calculator to solve the system of linear equations.

23. 0.2x + 0.4y = 4 24. −1.6x − 3.2y = −24

−0.6x + 0.6y = −3 2.6x + 2.6y = 26

25. −7x + 6y = 0 26. 4x − y = 1.5

0.5x + y = 2 2x + y = 1.5

27. MODELING WITH MATHEMATICS You have

40 minutes to exercise at the gym, and you want to

burn 300 calories total using both machines. How

much time should you spend on each machine?

(See Example 3.)

Elliptical Trainer

8 calories per minute

Stationary Bike

6 calories per minute

28. MODELING WITH MATHEMATICS

You sell small and large candles

at a craft fair. You collect $144

selling a total of 28 candles.

How many of each type of candle

did you sell?

29. MATHEMATICAL CONNECTIONS Write a linear

equation that represents the area and a linear equation

that represents the perimeter of the rectangle. Solve

the system of linear equations by graphing. Interpret

your solution.

6 cm

(3x − 3) cm

30. THOUGHT PROVOKING Your friend’s bank account

balance (in dollars) is represented by the equation

y = 25x + 250, where x is the number of months.

Graph this equation. After 6 months, you want to

have the same account balance as your friend. Write a

linear equation that represents your account balance.

Interpret the slope and y-intercept of the line that

represents your account balance.

31. COMPARING METHODS Consider the equation

x + 2 = 3x − 4.

a. Solve the equation using algebra.

b. Solve the system of linear equations y = x + 2

and y = 3x − 4 by graphing.

c. How is the linear system and the solution in part

(b) related to the original equation and the solution

in part (a)?

32. HOW DO YOU SEE IT? A teacher is purchasing

binders for students. The graph shows the total costs

of ordering x binders from three different companies.

1500

50

75

100

125

150

20 25 30 35

Co

st (

do

llars

)

40 45 50 x

y

Number of binders

Buying Binders

Company A

Company B

Company C

a. For what numbers of binders are the costs the

same at two different companies? Explain.

b. How do your answers in part (a) relate to systems

of linear equations?

33. MAKING AN ARGUMENT You and a friend are going

hiking but start at different locations. You start at the

trailhead and walk 5 miles per hour. Your friend starts

3 miles from the trailhead and walks 3 miles per hour.

you

your friend

a. Write and graph a system of linear equations that

represents this situation.

b. Your friend says that after an hour of hiking you

will both be at the same location on the trail. Is

your friend correct? Use the graph from part (a) to

explain your answer.

$6each $4

each

Section 5.2 Solving Systems of Linear Equations by Substitution 225

5.2 Solving Systems of Linear Equations by Substitution

Using Substitution to Solve Systems

Work with a partner. Solve each system of linear equations using two methods.

Method 1 Solve for x fi rst.Solve for x in one of the equations. Substitute the expression for x into the other

equation to fi nd y. Then substitute the value of y into one of the original equations

to fi nd x.

Method 2 Solve for y fi rst.Solve for y in one of the equations. Substitute the expression for y into the other

equation to fi nd x. Then substitute the value of x into one of the original equations

to fi nd y.

Is the solution the same using both methods? Explain which method you would prefer

to use for each system.

a. x + y = −7 b. x − 6y = −11 c. 4x + y = −1

−5x + y = 5 3x + 2y = 7 3x − 5y = −18

Essential QuestionEssential Question How can you use substitution to solve a system


Writing and Solving a System of Equations

Work with a partner.

a. Write a random ordered pair with integer

coordinates. One way to do this is to use

a graphing calculator. The ordered pair

generated at the right is (−2, −3).

b. Write a system of linear equations that has

your ordered pair as its solution.

c. Exchange systems with your partner and use

one of the methods from Exploration 1 to

solve the system. Explain your choice

of method.

Communicate Your AnswerCommunicate Your Answer 3. How can you use substitution to solve a system of linear equations?

4. Use one of the methods from Exploration 1 to solve each system of linear

equations. Explain your choice of method. Check your solutions.

a. x + 2y = −7 b. x − 2y = −6 c. −3x + 2y = −10

2x − y = −9 2x + y = −2 −2x + y = −6

d. 3x + 2y = 13 e. 3x − 2y = 9 f. 3x − y = −6

x − 3y = −3 −x − 3y = 8 4x + 5y = 11

USING PRECISE MATHEMATICAL LANGUAGE

To be profi cient in math, you need to communicate precisely with others.

randInt(-5‚5‚2){-2 -3}

Choose tworandom integersbetween −5 and 5.

A.2.IA.5.C



5.2 Lesson What You Will LearnWhat You Will Learn Solve systems of linear equations by substitution.


Solving Linear Systems by SubstitutionPrevioussystem of linear equationssolution of a system of linear equations


Core Core ConceptConceptSolving a System of Linear Equations by SubstitutionStep 1 Solve one of the equations for one of the variables.

Step 2 Substitute the expression from Step 1 into the other equation and

solve for the other variable.

Step 3 Substitute the value from Step 2 into one of the original equations

and solve.

Solving a System of Linear Equations by Substitution

Solve the system of linear equations by substitution.

y = −2x − 9 Equation 1


SOLUTION

Step 1 Equation 1 is already solved for y.

Step 2 Substitute −2x − 9 for y in Equation 2 and solve for x.


6x − 5(−2x − 9) = −19 Substitute −2x − 9 for y.

6x + 10x + 45 = −19 Distributive Property

16x + 45 = −19 Combine like terms.

16x = −64 Subtract 45 from each side.

x = −4 Divide each side by 16.

Step 3 Substitute −4 for x in Equation 1 and solve for y.


= −2(−4) − 9 Substitute −4 for x.

= 8 − 9 Multiply.

= −1 Subtract.

The solution is (−4, −1).


Solve the system of linear equations by substitution. Check your solution.

1. y = 3x + 14 2. 3x + 2y = 0 3. x = 6y − 7

y = −4x y = 1 —

2 x − 1 4x + y = −3

Another way to solve a system of linear equations is to use substitution.

Check

Equation 1

y = −2x − 9

−1 =?

−2(−4) − 9

−1 = −1 ✓

Equation 2

6x − 5y = −19

6(−4) − 5(−1) =?

−19

−19 = −19 ✓


Solving a System of Linear Equations by Substitution

Solve the system of linear equations by substitution.

−x + y = 3 Equation 1

3x + y = −1 Equation 2

SOLUTION

Step 1 Solve for y in Equation 1.

y = x + 3 Revised Equation 1

Step 2 Substitute x + 3 for y in Equation 2 and solve for x.

3x + y = −1 Equation 2

3x + (x + 3) = −1 Substitute x + 3 for y.

4x + 3 = −1 Combine like terms.

4x = −4 Subtract 3 from each side.


Step 3 Substitute −1 for x in Equation 1 and solve for y.

−x + y = 3 Equation 1

−(−1) + y = 3 Substitute −1 for x.

y = 2 Subtract 1 from each side.

The solution is (−1, 2).

Algebraic Check

Equation 1

−x + y = 3

−(−1) + 2 =?

3

3 = 3 ✓

Equation 2

3x + y = −1

3(−1) + 2 =?

−1

−1 = −1 ✓

Graphical Check

−5

−2

4

4

y = x + 3

4

y = −3x − 1

IntersectionX=-1 Y=2



4. x + y = −2 5. −x + y = −4

−3x + y = 6 4x − y = 10

6. 2x − y = −5 7. x − 2y = 7

3x − y = 1 3x − 2y = 3

ANOTHER WAYYou could also begin by solving for x in Equation 1, solving for y in Equation 2, or solving for x in Equation 2.



A drama club earns $1040 from a production. An adult

ticket costs twice as much as a student ticket. Write a

system of linear equations that represents this situation.

What is the price of each type of ticket?

SOLUTION

1. Understand the Problem You know the amount earned, the total numbers of adult

and student tickets sold, and the relationship between the price of an adult ticket

and the price of a student ticket. You are asked to write a system of linear equations

that represents the situation and fi nd the price of each type of ticket.

2. Make a Plan Use a verbal model to write a system of linear equations that

represents the problem. Then solve the system of linear equations.


Words 64 ⋅ Adult ticket

price + 132 ⋅

Student

ticket price = 1040

Adult ticket

price = 2 ⋅

Student

ticket price

Variables Let x be the price (in dollars) of an adult ticket and let y be the

price (in dollars) of a student ticket.


x = 2y Equation 2

Step 1 Equation 2 is already solved for x.

Step 2 Substitute 2y for x in Equation 1 and solve for y.

64x + 132y = 1040 Equation 1

64(2y) + 132y = 1040 Substitute 2y for x.

260y = 1040 Simplify.

y = 4 Simplify.

Step 3 Substitute 4 for y in Equation 2 and solve for x.

x = 2y Equation 2

x = 2(4) Substitute 4 for y.

x = 8 Simplify.

The solution is (8, 4). So, an adult ticket costs $8 and a student ticket costs $4.

4. Look Back To check that your solution is correct, substitute the values of x and

y into both of the original equations and simplify.

64(8) + 132(4) = 1040 8 = 2(4)

1040 = 1040 ✓ 8 = 8 ✓


8. There are a total of 64 students in a drama club and a yearbook club. The drama

club has 10 more students than the yearbook club. Write a system of linear

equations that represents this situation. How many students are in each club?


STUDY TIPYou can use either of the original equations to solve for x. However, using Equation 2 requires fewer calculations.

A

t

s

W

S

1

2

3

Tickets sold

Type Number

adult 64

student 132



Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3−8, tell which equation you would choose to solve for one of the variables. Explain.

3. x + 4y = 30 4. 3x − y = 0

x − 2y = 0 2x + y = −10

5. 5x + 3y = 11 6. 3x − 2y = 19

5x − y = 5 x + y = 8

7. x − y = −3 8. 3x + 5y = 25

4x + 3y = −5 x − 2y = −6

In Exercises 9–16, solve the sytem of linear equations by substitution. Check your solution. (See Examples 1 and 2.)

9. x = 17 − 4y 10. 6x − 9 = y

y = x − 2 y = −3x

11. x = 16 − 4y 12. −5x + 3y = 51

3x + 4y = 8 y = 10x − 8

13. 2x = 12 14. 2x − y = 23

x − 5y = −29 x − 9 = −1

15. 5x + 2y = 9 16. 11x − 7y = −14

x + y = −3 x − 2y = −4

17. ERROR ANALYSIS Describe and correct the error in

solving for one of the variables in the linear system

8x + 2y = −12 and 5x − y = 4.

Step 1 5x − y = 4

−y = −5x + 4

y = 5x − 4

Step 2 5x − (5x − 4) = 4

5x − 5x + 4 = 4

4 = 4

✗



4x + 2y = 6 and 3x + y = 9.

Step 1 3x + y = 9

y = 9 − 3x

Step 2 4x + 2(9 − 3x) = 6

4x + 18 − 6x = 6

−2x = −12

x = 6

Step 3 3x + y = 9

3x + 6 = 9

3x = 3

x = 1

✗

19. MODELING WITH MATHEMATICS A farmer plants

corn and wheat on a 180-acre farm. The farmer wants

to plant three times as many acres of corn as wheat.

Write a system of linear equations that represents this

situation. How many acres of each crop should the

farmer plant? (See Example 3.)

20. MODELING WITH MATHEMATICS A company that

offers tubing trips down a river rents tubes for a

person to use and “cooler” tubes to carry food and

water. A group spends $270 to rent a total of 15 tubes.

Write a system of linear equations that represents this

situation. How many of each type of tube does the

group rent?

1. WRITING Describe how to solve a system of linear equations by substitution.

2. NUMBER SENSE When solving a system of linear equations by substitution, how do you decide

which variable to solve for in Step 1?



Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyFind the sum or difference. (Skills Review Handbook)

36. (x − 4) + (2x − 7) 37. (5y − 12) + (−5y − 1)

38. (t − 8) − (t + 15) 39. (6d + 2) − (3d − 3)

40. 4(m + 2) + 3(6m − 4) 41. 2(5v + 6) − 6(−9v + 2)


In Exercises 21–24, write a system of linear equations that has the ordered pair as its solution.

21. (3, 5) 22. (−2, 8)

23. (−4, −12) 24. (15, −25)

25. PROBLEM SOLVING A math test is worth 100 points

and has 38 problems. Each problem is worth either

5 points or 2 points. How many problems of each

point value are on the test?

26. PROBLEM SOLVING An investor owns shares of

Stock A and Stock B. The investor owns a total of

200 shares with a total value of $4000. How many

shares of each stock does the investor own?

Stock Price

A $9.50

B $27.00

MATHEMATICAL CONNECTIONS In Exercises 27 and 28, (a) write an equation that represents the sum of the angle measures of the triangle and (b) use your equation and the equation shown to fi nd the values of x and y.

27.

x + 2 = 3y

x°

y°

28.

x°

y °(y − 18)°

3x − 5y = −22

29. REASONING Find the values of a and b so that the

solution of the linear system is (−9, 1).

ax + by = −31 Equation 1

ax − by = −41 Equation 2

30. MAKING AN ARGUMENT Your friend says that given

a linear system with an equation of a horizontal line

and an equation of a vertical line, you cannot solve

the system by substitution. Is your friend correct?

Explain.

31. OPEN-ENDED Write a system of linear equations in

which (3, −5) is a solution of Equation 1 but not a

solution of Equation 2, and (−1, 7) is a solution of

the system.

32. HOW DO YOU SEE IT? The graphs of two linear

equations are shown.

2 4 6 x

2

4

6

y y = x + 1

y = 6 − x14

a. At what point do the lines appear to intersect?

b. Could you solve a system of linear equations by

substitution to check your answer in part (a)?

Explain.

33. REPEATED REASONING A radio station plays a total

of 272 pop, rock, and hip-hop songs during a day. The

number of pop songs is 3 times the number of rock

songs. The number of hip-hop songs is 32 more than

the number of rock songs. How many of each type of

song does the radio station play?

34. THOUGHT PROVOKING You have $2.65 in coins.

Write a system of equations that represents this

situation. Use variables to represent the number of

each type of coin.

35. NUMBER SENSE The sum of the digits of a

two-digit number is 11. When the digits are reversed,

the number increases by 27. Find the original number.

Section 5.3 Solving Systems of Linear Equations by Elimination 231

5.3 Solving Systems of Linear Equations by Elimination

Writing and Solving a System of Equations

Work with a partner. You purchase a drink and a sandwich for $4.50. Your friend

purchases a drink and fi ve sandwiches for $16.50. You want to determine the price of

a drink and the price of a sandwich.

a. Let x represent the price (in dollars) of one drink. Let y represent the price

(in dollars) of one sandwich. Write a system of equations for the situation. Use

the following verbal model.

Number

of drinks ⋅ Price

per drink +

Number of

sandwiches ⋅ Price per

sandwich =

Total

price

Label one of the equations Equation 1 and the other equation Equation 2.

b. Subtract Equation 1 from Equation 2. Explain how you can use the result to solve

the system of equations. Then fi nd and interpret the solution.

Essential QuestionEssential Question How can you use elimination to solve a system


Using Elimination to Solve a System


2x + y = 7 Equation 1

x + 5y = 17 Equation 2

a. Can you eliminate a variable by adding or subtracting the equations as they are?

If not, what do you need to do to one or both equations so that you can?

b. Solve the system individually. Then exchange solutions with your partner and

compare and check the solutions.

Communicate Your AnswerCommunicate Your Answer 4. How can you use elimination to solve a system of linear equations?

5. When can you add or subtract the equations in a system to solve the system?

When do you have to multiply fi rst? Justify your answers with examples.

6. In Exploration 3, why can you multiply an equation in the system by a constant

and not change the solution of the system? Explain your reasoning.

Using Elimination to Solve Systems

Work with a partner. Solve each system of linear equations using two methods.

Method 1 Subtract. Subtract Equation 2 from Equation 1. Then use the result to

solve the system.

Method 2 Add. Add the two equations. Then use the result to solve the system.

Is the solution the same using both methods? Which method do you prefer?

a. 3x − y = 6 b. 2x + y = 6 c. x − 2y = −7

3x + y = 0 2x − y = 2 x + 2y = 5

USING PROBLEM-SOLVING STRATEGIES

To be profi cient in math, you need to monitor and evaluate your progress and change course using a different solution method, if necessary.

A.2.IA.5.C



5.3 Lesson What You Will LearnWhat You Will Learn Solve systems of linear equations by elimination.


Solving Linear Systems by EliminationPreviouscoeffi cient


Core Core ConceptConceptSolving a System of Linear Equations by EliminationStep 1 Multiply, if necessary, one or both equations by a constant so at least one

pair of like terms has the same or opposite coeffi cients.

Step 2 Add or subtract the equations to eliminate one of the variables.

Step 3 Solve the resulting equation.

Step 4 Substitute the value from Step 3 into one of the original equations and

solve for the other variable.

Solving a System of Linear Equations by Elimination

Solve the system of linear equations by elimination.

3x + 2y = 4 Equation 1


SOLUTION

Step 1 Because the coeffi cients of the y-terms are opposites, you do not need to

multiply either equation by a constant.

Step 2 Add the equations.



6x = 0 Add the equations.

Step 3 Solve for x.

6x = 0 Resulting equation from Step 2

x = 0 Divide each side by 6.

Step 4 Substitute 0 for x in one of the original equations and solve for y.


3(0) + 2y = 4 Substitute 0 for x.

y = 2 Solve for y.

The solution is (0, 2).

Check

Equation 1

3x + 2y = 4

3(0) + 2(2) =?

4

4 = 4 ✓Equation 2

3x − 2y = −4

3(0) − 2(2) =?

−4

−4 = −4 ✓

You can use elimination to solve a system of equations because replacing one

equation in the system with the sum of that equation and a multiple of the other

produces a system that has the same solution. Here is why.

Consider System 1. In this system, a and c are algebraic expressions, and b and d are

constants. Begin by multiplying each side of Equation 2 by a constant k. By the

Multiplication Property of Equality, kc = kd. You can rewrite Equation 1 as

Equation 3 by adding kc on the left and kd on the right. You can rewrite Equation 3 as

Equation 1 by subtracting kc on the left and kd on the right. Because you can rewrite

either system as the other, System 1 and System 2 have the same solution.

System 1

a = b Equation 1c = d Equation 2

System 2

a + kc = b + kd Equation 3c = d Equation 2


Solving a System of Linear Equations by Elimination

Solve the system of linear equations by elimination.

−10x + 3y = 1 Equation 1

−5x − 6y = 23 Equation 2

SOLUTION

Step 1 Multiply Equation 2 by −2 so that the coeffi cients of the x-terms

are opposites.

−10x + 3y = 1 −10x + 3y = 1 Equation 1

−5x − 6y = 23 Multiply by −2. 10x + 12y = −46 Revised Equation 2


− 10x + 3y = 1 Equation 1

10x + 12y = −46 Revised Equation 2

15y = −45 Add the equations.

Step 3 Solve for y.

15y = −45 Resulting equation from Step 2

y = −3 Divide each side by 15.

Step 4 Substitute −3 for y in one of the original equations and solve for x.

−5x − 6y = 23 Equation 2

−5x − 6(−3) = 23 Substitute −3 for y.

−5x + 18 = 23 Multiply.

−5x = 5 Subtract 18 from each side.

x = −1 Divide each side by −5.

The solution is (−1, −3).

Monitoring Progress Help in English and Spanish at BigIdeasMath.com

Solve the system of linear equations by elimination. Check your solution.

1. 3x + 2y = 7 2. x − 3y = 24 3. x + 4y = 22

−3x + 4y = 5 3x + y = 12 4x + y = 13

ANOTHER WAYTo use subtraction toeliminate one of the variables, multiply Equation 2 by 2 and then subtract the equations.

− 10x + 3y = 1

−(−10x − 12y = 46)

15y = −45

Methods for Solving Systems of Linear Equations

Concept SummaryConcept Summary

Method When to Use

Graphing (Lesson 5.1) To estimate solutions

Substitution

(Lesson 5.2)When one of the variables in one of the

equations has a coeffi cient of 1 or −1

Elimination

(Lesson 5.3)When at least one pair of like terms has the

same or opposite coeffi cients

Elimination (Multiply First)

(Lesson 5.3)When one of the variables cannot be eliminated

by adding or subtracting the equations

Check

10

−10

−10

10

IntersectionX=-1 Y=-3

10

Equation 110

3

Equation 2



The graph represents the average salaries of classroom teachers in two school districts.

During what year were the average salaries in the two districts equal? What was the

average salary in both districts in that year?

SOLUTION

1. Understand the Problem You know two points on each line in the graph. You

are asked to determine the year in which the average salaries were equal and then

determine the average salary in that year.

2. Make a Plan Use the points in the graph to write a system of linear equations.

Then solve the system of linear equations.

3. Solve the Problem Find the slope of each line.

District A: m = 60 − 25

— 25 − 5

= 35

— 20

= 7 —

4 District B: m =

55 − 25 —

25 − 0 =

30 —

25 =

6 —

5

Use each slope and a point on each line to write equations of the lines.

District A District B

y − y1 = m(x − x1) Write the point-slope form. y − y1 = m(x − x1)

y − 25 = 7 —

4 (x − 5) Substitute for m, x1, and y1. y − 25 =

6 —

5 (x − 0)

−7x + 4y = 65 Write in standard form. −6x + 5y = 125

System −7x + 4y = 65 Equation 1


Step 1 Multiply Equation 1 by −5. Multiply Equation 2 by 4.

−7x + 4y = 65 Multiply by −5. 35x − 20y = −325 Revised Equation 1

−6x + 5y = 125 Multiply by 4. −24x + 20y = 500 Revised Equation 2


35x − 20y = −325 Revised Equation 1

−24x + 20y = 500 Revised Equation 2

11x = 175 Add the equations.

Step 3 Solving the equation 11x = 175 gives x = 175

— 11

≈ 15.9.

Step 4 Substitute 175

— 11

for x in one of the original equations and solve for y.


−7 ( 175 —

11 ) + 4y = 65 Substitute 175

— 11 for x.

y ≈ 44 Solve for y.

The solution is about (15.9, 44). Because x ≈ 15.9 corresponds to the

year 2000, the average salary in both districts was about $44,000 in 2000.

4. Look Back Using the graph, the point of intersection appears to be about (15, 45).

So, the solution of (15.9, 44) is reasonable.


4. Write and solve a system of linear equations represented by the graph at the left.


STUDY TIPIn Example 3, both equations are multiplied by a constant so that the coeffi cients of the y-terms are opposites.

00

20

40

10 20 30 x

y

Years since 1985

60

80

Classroom Teacher

Ave

rag

e sa

lary

(th

ou

san

ds

of

do

llars

) District ADistrict B

(25, 55)

(25, 60)

(5, 25)

(0, 25)

00

4

8

4 8 12 16 x

y

12

16

(2, 9)

(14, 15)

(4, 4)

(8, 16)Line B Line A



Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3−10, solve the system of linear equations by elimination. Check your solution. (See Example 1.)

3. x + 2y = 13 4. 9x + y = 2

−x + y = 5 −4x − y = −17

5. 5x + 6y = 50 6. −x + y = 4

x − 6y = −26 x + 3y = 4

7. −3x − 5y = −7 8. 4x − 9y = −21

−4x + 5y = 14 −4x − 3y = 9

9. −y − 10 = 6x 10. 3x − 30 = y

5x + y = −10 7y − 6 = 3x

In Exercises 11–18, solve the system of linear equations by elimination. Check your solution. (See Examples 2 and 3.)

11. x + y = 2 12. 8x − 5y = 11

2x + 7y = 9 4x − 3y = 5

13. 11x − 20y = 28 14. 10x − 9y = 46

3x + 4y = 36 −2x + 3y = 10

15. 4x − 3y = 8 16. −2x − 5y = 9

5x − 2y = −11 3x + 11y = 4

17. 9x + 2y = 39 18. 12x − 7y = −2

6x + 13y = −9 8x + 11y = 30



5x − 7y = 16 and x + 7y = 8.

5x − 7y = 16

x + 7y = 8

4x = 24

x = 6

✗



4x + 3y = 8 and x − 2y = −13.

21. MODELING WITH MATHEMATICS A service center

charges a fee of x dollars for an oil change plus

y dollars per quart of oil used. A sample of its sales

record is shown. Write a system of linear equations

that represents this situation. Find the fee and cost

per quart of oil.

A B

2

1

34

Customer Oil Tank Size(quarts)

TotalCost

A 5B 7

C

$22.45$25.45

22. MODELING WITH MATHEMATICS The graph represents

the average salaries of

high school principals

in two states. During

what year were the

average salaries in

the two states equal?

What was the average

salary in both states

in that year?

1. OPEN-ENDED Give an example of a system of linear equations that can be solved by fi rst adding

the equations to eliminate one variable.

2. WRITING Explain how to solve the system of linear equations 2x − 3y = −4 Equation 1−5x + 9y = 7 Equation 2by elimination.


4x + 3y = 8 4x + 3y = 8

x − 2y = −13 Multiply by −4. −4x + 8y = −13

11y = −5

y = −5

— 11

✗

00

40

80

10 20 30 x

y

Years since 1985

120

160

High School Principal

Ave

rag

e sa

lary

(th

ou

san

ds

of

do

llars

)

State AState B

(22, 100)

(25, 98)(0, 43)

(6, 48)


Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the equation. Determine whether the equation has one solution, no solution, or infi nitely many solutions. (Section 1.3)

36. 5d − 8 = 1 + 5d 37. 9 + 4t = 12 − 4t

38. 3n + 2 = 2(n − 3) 39. −3(4 − 2v) = 6v − 12

Write an equation of the line that passes through the given point and is parallel to the given line. (Section 4.4)

40. (4, −1); y = −2x + 7 41. (0, 6); y = 5x − 3 42. (−5, −2); y = 2 —

3 x + 1


In Exercises 23–26, solve the system of linear equations using any method. Explain why you chose the method.

23. 3x + 2y = 4 24. −6y + 2 = −4x

2y = 8 − 5x y − 2 = x

25. y − x = 2 26. 3x + y = 1 —

3

y = − 1 — 4 x + 7 2x − 3y =

8 —

3

27. WRITING For what values of a can you solve the

linear system ax + 3y = 2 and 4x + 5y = 6 by

elimination without multiplying fi rst? Explain.

28. HOW DO YOU SEE IT? The circle graph shows the

results of a survey in which 50 students were asked

about their favorite meal.

Favorite Meal

Dinner25

Breakfast

Lunch

a. Estimate the numbers of students who chose

breakfast and lunch.

b. The number of students who chose lunch was

5 more than the number of students who chose

breakfast. Write a system of linear equations that

represents the numbers of students who chose

breakfast and lunch.

c. Explain how you can solve the linear system in

part (b) to check your answers in part (a).

29. MAKING AN ARGUMENT Your friend says that any

system of equations that can be solved by elimination

can be solved by substitution in an equal or fewer

number of steps. Is your friend correct? Explain.

30. THOUGHT PROVOKING Write a system of linear

equations that can be added to eliminate a variable

or subtracted to eliminate a variable.

31. MATHEMATICAL CONNECTIONS A rectangle has a

perimeter of 18 inches. A new rectangle is formed

by doubling the width w and tripling the lengthℓ,

as shown. The new rectangle has a perimeter P

of 46 inches.

P = 46 in. 2w

3

a. Write and solve a system of linear equations to

fi nd the length and width of the original rectangle.

b. Find the length and width of the new rectangle.

32. CRITICAL THINKING Refer to the discussion of

System 1 and System 2 on page 232. Without solving,

explain why the two systems shown have the

same solution.

System 1 System 2

3x − 2y = 8 Equation 1 5x = 20 Equation 3x + y = 6 Equation 2 x + y = 6 Equation 2

33. PROBLEM SOLVING You are making 6 quarts of

fruit punch for a party. You have bottles of 100% fruit

juice and 20% fruit juice. How many quarts of each

type of juice should you mix to make 6 quarts of 80%

fruit juice?

34. PROBLEM SOLVING A motorboat takes 40 minutes

to travel 20 miles downstream. The return trip takes

60 minutes. What is the speed of the current?

35. CRITICAL THINKING Solve for x, y, and z in the

system of equations. Explain your steps.

x + 7y + 3z = 29 Equation 1 3z + x − 2y = −7 Equation 2 5y = 10 − 2x Equation 3

Section 5.4 Solving Special Systems of Linear Equations 237

5.4 Solving Special Systems of Linear Equations

Using a Table to Solve a System

Work with a partner. You invest $450 for equipment to make skateboards. The

materials for each skateboard cost $20. You sell each skateboard for $20.

a. Write the cost and revenue equations. Then copy and complete the table for your

cost C and your revenue R.

b. When will your company break even? What is wrong?

Essential QuestionEssential Question Can a system of linear equations have no

solution or infi nitely many solutions?

Writing and Analyzing a System

Work with a partner. A necklace and matching bracelet have two types of beads.

The necklace has 40 small beads and 6 large beads and weighs 10 grams. The bracelet

has 20 small beads and 3 large beads and weighs 5 grams. The threads holding the

beads have no signifi cant weight.

a. Write a system of linear equations that represents the situation. Let x be the weight

(in grams) of a small bead and let y be the weight (in grams) of a large bead.

b. Graph the system in the coordinate plane shown.

What do you notice about the two lines?

c. Can you fi nd the weight of each type of bead?

Explain your reasoning.

Communicate Your AnswerCommunicate Your Answer 3. Can a system of linear equations have no solution or infi nitely many solutions?

Give examples to support your answers.

4. Does the system of linear equations represented by each graph have no solution,

one solution, or infi nitely many solutions? Explain.

a.

x

y

4

1

42−1

y = x + 2

x + y = 2

b.

x

y

3

6

42

3

y = x + 2

−x + y = 1

c.

x

y

3

1

6

42

3

y = x + 2

−2x + 2y = 4

x (skateboards) 0 1 2 3 4 5 6 7 8 9 10

C (dollars)

R (dollars)

APPLYING MATHEMATICSTo be profi cient in math, you need to interpret mathematical results in real-life contexts.

x

y

1

1.5

2

0.5

00.2 0.3 0.40.10

A.2.IA.3.FA.5.C



5.4 Lesson What You Will LearnWhat You Will Learn Determine the numbers of solutions of linear systems.

Use linear systems to solve real-life problems.

The Numbers of Solutions of Linear SystemsPreviousparallel


Core Core ConceptConceptSolutions of Systems of Linear EquationsA system of linear equations can have one solution, no solution, or infi nitely many solutions.

One solution No solution Infi nitely many solutions

x

y

x

y

x

y

The lines intersect. The lines are parallel. The lines are the same.

Solving a System: No Solution

Solve the system of linear equations.

y = 2x + 1 Equation 1


SOLUTION

Method 1 Solve by graphing.

Graph each equation.

The lines have the same slope and different

y-intercepts. So, the lines are parallel.

Because parallel lines do not intersect,

there is no point that is a solution

of both equations.

So, the system of linear equations

has no solution.

Method 2 Solve by substitution.

Substitute 2x − 5 for y in Equation 1.

y = 2x + 1 Equation 1

2x − 5 = 2x + 1 Substitute 2x − 5 for y.

−5 = 1 ✗ Subtract 2x from each side.

The equation −5 = 1 is never true. So, the system of linear equations

has no solution.

ANOTHER WAYYou can solve some linear systems by inspection. In Example 1, notice you can rewrite the system as

–2x + y = 1–2x + y = –5.

This system has no solution because –2x + y cannot be equal to both 1 and –5.

STUDY TIPA linear system with no solution is called an inconsistent system.

x

y

2

−2

−4

41−2

1

2

1

2

2

−22

y = 2x + 1

x44

y = 2x − 5


Solving a System: Infi nitely Many Solutions


−2x + y = 3 Equation 1


SOLUTION

Solve by elimination.

Step 1 Multiply Equation 1 by −2.

−2x + y = 3 Multiply by −2. 4x − 2y = −6 Revised Equation 1

−4x + 2y = 6 −4x + 2y = 6 Equation 2




0 = 0 Add the equations.

The equation 0 = 0 is always true. So, the solutions are all the points on the line

−2x + y = 3. The system of linear equations has infi nitely many solutions.



1. x + y = 3 2. y = −x + 3

2x + 2y = 6 2x + 2y = 4

ANOTHER WAYYou can also solve the linear system by graphing. The lines have the same slope and the same y-intercept. So, the lines are the same, which means all points on the line are solutions of both equations.

STUDY TIPA linear system with infi nitely many solutions is called a consistent dependent system.



An athletic director is comparing the costs of renting two banquet halls for an awards

banquet. Write a system of linear equations that represents this situation. If the cost

patterns continue, will the cost of Hall A ever equal the cost of Hall B?

SOLUTION

Words Total cost = Cost per hour ⋅ Number of hours + Initial cost

Variables Let y be the cost (in dollars) and let x be the number of hours.

System y = 100x + 75 Equation 1 - Cost of Hall A

y = 100x + 100 Equation 2 - Cost of Hall B

The equations are in slope-intercept form. The graphs of the equations have the same

slope but different y-intercepts. There is no solution because the lines are parallel.

So, the cost of Hall A will never equal the cost of Hall B.


3. WHAT IF? What happens to the solution in Example 3 when the cost per hour for

Hall A is $125?

Cost (dollars)

Hours Hall A Hall B

0 75 100

1 175 200

2 275 300

3 375 400



The perimeter of the trapezoidal piece of land is 48 kilometers. The perimeter of

the rectangular piece of land is 144 kilometers. Write and solve a system of linear

equations to fi nd the values of x and y.

SOLUTION

1. Understand the Problem You know the perimeter of each piece of land and the

side lengths in terms of x or y. You are asked to write and solve a system of linear

equations to fi nd the values of x and y.

2. Make a Plan Use the fi gures and the defi nition of perimeter to write a

system of linear equations that represents the problem. Then solve the system

of linear equations.


Perimeter of trapezoid Perimeter of rectangle

2x + 4x + 6y + 6y = 48 9x + 9x + 18y + 18y = 144

6x + 12y = 48 Equation 1 18x + 36y = 144 Equation 2


18x + 36y = 144 Equation 2

Method 1 Solve by graphing.

Graph each equation.

The lines have the same slope and the same

y-intercept. So, the lines are the same.

In this context, x and y must be positive.

Because the lines are the same, all the points

on the line in Quadrant I are solutions of

both equations.

So, the system of linear equations has infi nitely many solutions.

Method 2 Solve by elimination.

Multiply Equation 1 by −3 and add the equations.

6x + 12y = 48 Multiply by −3. −18x − 36y = −144 Revised Equation 1

18x + 36y = 144 18x + 36y = 144 Equation 2

0 = 0 Add the equations.

The equation 0 = 0 is always true. In this context, x and y must be positive.

So, the solutions are all the points on the line 6x + 12y = 48 in Quadrant I.

The system of linear equations has infi nitely many solutions.

4. Look Back Choose a few of the ordered pairs (x, y) that are solutions of Equation 1.

You should fi nd that no matter which ordered pairs you choose, they will also be

solutions of Equation 2. So, infi nitely many solutions seems reasonable.


4. WHAT IF? What happens to the solution in Example 4 when the perimeter of the

trapezoidal piece of land is 96 kilometers? Explain.

4x

2x

6y 6y

18y

18y

9x9x

2 4 60 x

2

4

6

0

y

6x + 12y = 486x + 12y =

18x + 36y = 144



Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3−8, match the system of linear equations with its graph. Then determine whether the system has one solution, no solution, or infi nitely many solutions.

3. −x + y = 1 4. 2x − 2y = 4

x − y = 1 −x + y = −2

5. 2x + y = 4 6. x − y = 0

−4x − 2y = −8 5x − 2y = 6

7. −2x + 4y = 1 8. 5x + 3y = 17

3x − 6y = 9 x − 3y = −2

A.

x

y

2

4

2−1

B.

x

y

2

4

6

1 4−2

C.

x

y

2

−2

2 4−2

D.

x

y

2

−3

1 4

E.

x

y

2

−1 2 4

F.

x

y

2

−3

2−3

In Exercises 9–16, solve the system of linear equations. (See Examples 1 and 2.)

9. y = −2x − 4 10. y = −6x − 8

y = 2x − 4 y = −6x + 8

11. 3x − y = 6 12. −x + 2y = 7

−3x + y = −6 x − 2y = 7

13. 4x + 4y = −8 14. 15x − 5y = −20

−2x − 2y = 4 −3x + y = 4

15. 9x − 15y = 24 16. 3x − 2y = −5

6x − 10y = −16 4x + 5y = 47

In Exercises 17–22, use only the slopes and y-intercepts of the graphs of the equations to determine whether the system of linear equations has one solution, no solution, or infi nitely many solutions. Explain.

17. y = 7x + 13 18. y = −6x − 2

−21x + 3y = 39 12x + 2y = −6

19. 4x + 3y = 27 20. −7x + 7y = 1

4x − 3y = −27 2x − 2y = −18

21. −18x + 6y = 24 22. 2x − 2y = 16

3x − y = −2 3x − 6y = 30

ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in solving the system of linear equations.

23. −4x + y = 4

4x + y = 12

The lines do not intersect. So, the system

has no solution.

✗x

y

1

−3

2−2

24. y = 3x − 8

y = 3x − 12

The lines have the same slope. So, the system

has infi nitely many solutions.

✗

1. REASONING Is it possible for a system of linear equations to have exactly two solutions? Explain.

2. WRITING Compare the graph of a system of linear equations that has infi nitely many solutions and

the graph of a system of linear equations that has no solution.



Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencySolve the system of linear equations by graphing. (Section 5.1)

33. y = x − 6 34. y = 2x + 3 35. 2x + y = 6

y = −x + 10 y = −3x − 7 3x − 2y = 16


25. MODELING WITH MATHEMATICS The table shows the

distances two groups have traveled at different times

during a canoeing excursion. The groups continue

traveling at their current rates until they reach the

same destination. Let d be the distance traveled

and t be the time since 1 p.m. Write a system of linear

equations that represents this situation. Will Group B

catch up to Group A before reaching the destination?

Explain. (See Example 3.)

Distance Traveled (miles)

1 P.M. 2 P.M. 3 P.M. 4 P.M.

Group A 3 9 15 21

Group B 1 7 13 19

26. MODELING WITH MATHEMATICS A $6-bag of trail

mix contains 3 cups of dried fruit and 4 cups of

almonds. A $9-bag contains 4 1 —

2 cups of dried fruit and

6 cups of almonds. Write and solve a system of linear

equations to fi nd the price of 1 cup of dried fruit and

1 cup of almonds. (See Example 4.)

27. PROBLEM SOLVING A train travels from New York

City to Washington, D.C., and then back to New York

City. The table shows the number of tickets purchased

for each leg of the trip. The cost per ticket is the same

for each leg of the trip. Is there enough information to

determine the cost of one coach ticket? Explain.

DestinationCoach tickets

Business class

tickets

Money collected (dollars)

Washington, D.C. 150 80 22,860

New York City 170 100 27,280

28. THOUGHT PROVOKING Write a system of three

linear equations in two variables so that any two of

the equations have exactly one solution, but the entire

system of equations has no solution.

29. REASONING In a system of linear equations, one

equation has a slope of 2 and the other equation has

a slope of − 1 — 3 . How many solutions does the system

have? Explain.

30. HOW DO YOU SEE IT? The graph shows information

about the last leg of a 4 × 200-meter relay for

three relay teams. Team A’s runner ran about

7.8 meters per second, Team B’s runner ran about

7.8 meters per second, and Team C’s runner ran about

8.8 meters per second.

400

50

100

150

8 12 16 20 24 28

Dis

tan

ce (

met

ers)

x

y

Time (seconds)

Last Leg of 4 × 200-Meter Relay

Team CTeam B

Team A

a. Estimate the distance at which Team C’s runner

passed Team B’s runner.

b. If the race was longer, could Team C’s runner have

passed Team A’s runner? Explain.

c. If the race was longer, could Team B’s runner have

passed Team A’s runner? Explain.

31. ABSTRACT REASONING Consider the system of

linear equations y = ax + 4 and y = bx − 2, where

a and b are real numbers. Determine whether each

statement is always, sometimes, or never true. Explain

your reasoning.

a. The system has infi nitely many solutions.

b. The system has no solution.

c. When a < b, the system has one solution.

32. MAKING AN ARGUMENT One admission to an ice

skating rink costs x dollars, and renting a pair of

ice skates costs y dollars. Your friend says she can

determine the exact cost of one admission and one

skate rental. Is your friend correct? Explain.

Total

2Admissions3

38.00

Skate Rentals

$ Total

15 Admissions10

190.00

Skate Rentals

$

243243

5.1–5.4 What Did You Learn?

Study Errors

What Happens: You do not study the right material or you do not learn it well enough to remember it on a test without resources such as notes.

How to Avoid This Error: Take a practice test. Work with a study group. Discuss the topics on the test with your teacher. Do not try to learn a whole chapter’s worth of material in one night.

Core VocabularyCore Vocabularysystem of linear equations, p. 220 solution of a system of linear equations, p. 220

Core ConceptsCore ConceptsSection 5.1Solving a System of Linear Equations by Graphing, p. 221

Section 5.2Solving a System of Linear Equations by Substitution, p. 226

Section 5.3Solving a System of Linear Equations by Elimination, p. 232

Section 5.4Solutions of Systems of Linear Equations, p. 238

Mathematical ThinkingMathematical Thinking1. Describe the given information in Exercise 33 on page 230 and your plan for fi nding the solution.

2. Describe another real-life situation similar to Exercise 21 on page 235 and the mathematics that you

can apply to solve the problem.

3. What question(s) can you ask your friend to help her understand the error in the statement she made

in Exercise 32 on page 242?

Study Skills

Analyzing Your Errors


5.1–5.4 Quiz

Use the graph to solve the system of linear equations. Check your solution. (Section 5.1)

1. y = − 1 — 3 x + 2 2. y =

1 —

2 x − 1 3. y = 1

y = x − 2 y = 4x + 6 y = 2x + 1

x

y3

1

−142−2

x

y2

−2

−4

−2−4

x

y

2

−2

2−2

Solve the system of linear equations by substitution. Check your solution. (Section 5.2)

4. y = x − 4 5. 2y + x = −4 6. 3x − 5y = 13

−2x + y = 18 y − x = −5 x + 4y = 10

Solve the system of linear equations by elimination. Check your solution. (Section 5.3)

7. x + y = 4 8. x + 3y = 1 9. 2x − 3y = −5

−3x − y = −8 5x + 6y = 14 5x + 2y = 16

Solve the system of linear equations. (Section 5.4)

10. x − y = 1 11. 6x + 2y = 16 12. 3x − 3y = −2

x − y = 6 2x − y = 2 −6x + 6y = 4

13. You plant a spruce tree that grows 4 inches per year and a hemlock tree

that grows 6 inches per year. The initial heights are shown. (Section 5.1)

a. Write a system of linear equations that represents this situation.

b. Solve the system by graphing. Interpret your solution.

14. It takes you 3 hours to drive to a concert 135 miles away. You drive

55 miles per hour on highways and 40 miles per hour on the rest of

the roads. (Section 5.1, Section 5.2, and Section 5.3)

a. How much time do you spend driving at each speed?

b. How many miles do you drive on highways? the rest of the roads?

15. In a football game, all of the home team’s points are from 7-point

touchdowns and 3-point fi eld goals. The team scores six times.

Write and solve a system of linear equations to fi nd the numbers

of touchdowns and fi eld goals that the home team scores.

(Section 5.1, Section 5.2, and Section 5.3)

sprucetree

hemlocktree

14 in.

8 in.

Section 5.5 Solving Equations by Graphing 245

Solving Equations by Graphing5.5

Solving an Equation by Graphing

Work with a partner. Solve 2x − 1 = − 1 — 2 x + 4 by graphing.

a. Use the left side to write a linear equation. Then use the right side to write

another linear equation.

b. Graph the two linear equations from

part (a). Find the x-value of the point of

intersection. Check that the x-value is the

solution of

2x − 1 = − 1 —

2 x + 4.

c. Explain why this “graphical method” works.

Essential QuestionEssential Question How can you use a system of linear

equations to solve an equation with variables on both sides?

Previously, you learned how to use algebra to solve equations with variables

on both sides. Another way is to use a system of linear equations.

Solving Equations Algebraically and Graphically

Work with a partner. Solve each equation using two methods.

Method 1 Use an algebraic method.

Method 2 Use a graphical method.

Is the solution the same using both methods?

a. 1 —

2 x + 4 = −

1 — 4 x + 1 b. 2

— 3 x + 4 =

1 —

3 x + 3

c. − 2 — 3 x − 1 =

1 —

3 x − 4 d. 4

— 5 x +

7 —

5 = 3x − 3

e. −x + 2.5 = 2x − 0.5 f. − 3x + 1.5 = x + 1.5

Communicate Your AnswerCommunicate Your Answer 3. How can you use a system of linear equations to solve an equation with

variables on both sides?

4. Compare the algebraic method and the graphical method for solving a

linear equation with variables on both sides. Describe the advantages and

disadvantages of each method.

SELECTING TOOLSTo be profi cient in math, you need to consider the available tools, which may include pencil and paper or a graphing calculator, when solving a mathematical problem.

x

y

2

4

6

2 4 6−2

−2

A.5.A



5.5 Lesson What You Will LearnWhat You Will Learn Solve linear equations by graphing.

Use linear equations to solve real-life problems.

Solving Linear Equations by GraphingYou can use a system of linear equations to solve an equation with variables

on both sides.

Previousabsolute value equation


Core Core ConceptConceptSolving Linear Equations by GraphingStep 1 To solve the equation ax + b = cx + d, write two linear equations.

ax + b = cx + d

and

Step 2 Graph the system of linear equations. The x-value of the solution

of the system of linear equations is the solution of the equation

ax + b = cx + d.

y = cx + dy = ax + b

Solving an Equation by Graphing

Solve −x + 1 = 2x − 5 by graphing. Check your solution.

SOLUTION

Step 1 Write a system of linear equations using each side of the original equation.

−x + 1 = 2x − 5

Step 2 Graph the system.

y = −x + 1 Equation 1


The graphs intersect at (2, −1).

So, the solution of the equation is x = 2.


Solve the equation by graphing. Check your solution.

1. 1 — 2 x − 3 = 2x 2. −4 + 9x = −3x + 2

Check

−x + 1 = 2x − 5

−(2) + 1 =?

2(2) − 5

−1 = −1 ✓

y = 2x − 5y = −x + 1

x

y1

1

y = −x + 1

y = 2x − 5

−1

−2

−4

(2, −1)




You are an ecologist studying the populations of

two types of fi sh in a lake. Use the information in

the table to predict when the populations of the

two types of fi sh will be equal.

SOLUTION

1. Understand the Problem You know the current population of each type of fi sh

and the annual change in each population. You are asked to determine when the

populations of the two types of fi sh will be equal.

2. Make a Plan Use a verbal model to write an equation that represents the problem.

Then solve the equation by graphing.


Words Type A Type B

Change

per year ⋅ Years + Current

population =

Change

per year ⋅ Years +

Current

population

Variable Let x be the number of years.

Equation −250x + 5750 = 400x + 2825

Solve the equation by graphing.


−250x + 5750 = 400x + 2825


y = −250x + 5750 Equation 1

y = 400x + 2825 Equation 2

The graphs appear to intersect at (4.5, 4625).

Check this solution in each equation of the

linear system, as shown.

So, the populations of the two types of

fi sh will be equal in 4.5 years.

4. Look Back To check that your solution

is correct, verify that x = 4.5 is the

solution of the original equation.

−250(4.5) + 5750 = 400(4.5) + 2825

4625 = 4625 ✓


3. WHAT IF? Type C has a current population of 3500 and grows by 500 each year.

Predict when the population of Type C will be equal to the population of Type A.

Check

y = −250x + 5750

4625 =?

−250(4.5) + 5750

4625 = 4625 ✓

y = 400x + 2825

4625 =?

400(4.5) + 2825

4625 = 4625 ✓Years

Fish

po

pu

lati

on

x

y

2500

3500

4500

5500

6500

02 3 4 510

(0, 5750)

(4.5, 4625)

(0, 2825)

750)

(4.5, 4625)

y = −250x + 5750

))y = 400x + 2825

Fish in a Lake

TypeCurrent

populationChange per year

A 5750 −250

B 2825 400

y = 400x + 2825y = −250x + 5750



Your family needs to rent a car for a week while on vacation. Company A charges

$3.25 per mile plus a fl at fee of $125 per week. Company B charges $3 per mile plus

a fl at fee of $150 per week. After how many miles of travel are the total costs the same

at both companies?

SOLUTION

1. Understand the Problem You know the costs of renting a car from two

companies. You are asked to determine how many miles of travel will result in the

same total costs at both companies.

2. Make a Plan Use a verbal model to write an equation that represents the problem.

Then solve the equation by graphing.


Words Company A Company B

Cost

per mile ⋅ Miles + Flat

fee =

Cost

per mile ⋅ Miles +

Flat

fee

Variable Let x be the number of miles traveled.

Equation 3.25x + 125 = 3x + 150

Solve the equation by graphing.


3.25x + 125 = 3x + 150

Step 2 Use a graphing calculator to graph the system.

00

600

150IntersectionX=100 Y=450

ction

y = 3.25x + 125

y = 3x + 150

Because the graphs intersect at (100, 450), the solution of the equation is x = 100.

So, the total costs are the same after 100 miles.

4. Look Back One way to check your solution is to solve the equation algebraically,

as shown.


4. WHAT IF? Company C charges $3.30 per mile plus a fl at fee of $115 per

week. After how many miles are the total costs the same at Company A and

Company C?

y = 3x + 150y = 3.25x + 125

Check

3.25x + 125 = 3x + 150

0.25x + 125 = 150

0.25x = 25

x = 100



Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3–10, use the graph to solve the equation. Check your solution.

3. x − 3 = 2 4. −3 = 4x + 1

x

y

−2

1

3

1 4

x

y

1

2−2

5. −2x + 3 = x 6. x − 4 = 3x

x

y

1

3

1 3−3

xy

−6

2−2

7. 2x + 1 = −x − 2 8. −5x − 1 = 3x − 1

x

y

−3

2

2−2

x

y

−12−2

9. −x − 1 = 1 —

3 x + 3 10. −

3 — 2 x − 2 = −4x + 3

x

y

2

4

1−2−4

x

y

2 4

−2

−4

−6

In Exercises 11−18, solve the equation by graphing. Check your solution. (See Example 1.)

11. x + 4 = −x 12. 4x = x + 3

13. x + 5 = −2x − 4 14. −2x + 6 = 5x − 1

15. 1 — 2 x − 2 = 9 − 5x 16. −5 +

1 —

4 x = 3x + 6

17. 5x − 7 = 2(x + 1) 18. −6(x + 4) = −3x − 6

In Exercises 19−24, solve the equation by graphing. Determine whether the equation has one solution, no solution, or infi nitely many solutions.

19. 3x − 1 = −x + 7 20. 5x − 4 = 5x + 1

21. −4(2 − x) = 4x − 8

22. −2x − 3 = 2(x − 2)

23. −x − 5 = − 1 — 3 (3x + 5)

24. 1 — 2 (8x + 3) = 4x +

3 —

2

In Exercises 25–28, write an equation that has the same solution as the linear system represented by the graph.

25.

x

y

−2

2

4

2 4

26.

x

y

2

4

2 4

27.

x

y

−4

2

2−4

28.

x

y

2

−2

4

1. REASONING The graphs of the equations y = 3x − 20 and y = −2x + 10 intersect at the

point (6, −2). Without solving, fi nd the solution of the equation 3x − 20 = −2x + 10.

2. WRITING Consider the equation ax + b = cx + d, where c = 0. Can you solve the equation by

graphing a system of linear equations? Explain.



Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyGraph the inequality. (Section 2.1)

40. y > 5 41. x ≤ −2 42. n ≥ 9 43. c < −6

Use the graphs of f and g to describe the transformation from the graph of f to the graph of g. (Section 3.7)

44. f(x) = x − 5; g(x) = f(x + 2) 45. f(x) = 6x; g(x) = −f(x)

46. f(x) = −2x + 1; g(x) = f(4x) 47. f(x) = 1 —

2 x − 2; g(x) = f(x − 1)


USING TOOLS In Exercises 29 and 30, use a graphing calculator to solve the equation.

29. 0.7x + 0.5 = −0.2x − 1.3

30. 2.1x + 0.6 = −1.4x + 6.9

31. MODELING WITH MATHEMATICS You need to hire

a catering company to serve meals to guests at a

wedding reception. Company A charges $500 plus

$20 per guest. Company B charges $800 plus

$16 per guest. For how many guests are the total costs

the same at both companies? (See Examples 2 and 3.)

32. MODELING WITH MATHEMATICS Your dog is

16 years old in dog years. Your cat is 28 years old in

cat years. For every human year, your dog ages by

7 dog years and your cat ages by 4 cat years. In how

many human years will both pets be the same age in

their respective types of years?

33. MODELING WITH MATHEMATICS You and a friend

race across a fi eld to a fence. Your friend has a

50-meter head start. The equations shown represent

you and your friend’s distances d (in meters) from the

fence t seconds after the race begins. Find the time at

which you catch up to your friend.

You: d = −5t + 200

Your friend: d = −3 1 —

3 t + 150

34. MAKING AN ARGUMENT The graphs of y = −x + 4

and y = 2x − 8 intersect at the point (4, 0). So,

your friend says the solution of the equation

−x + 4 = 2x − 8 is (4, 0). Is your friend correct?

Explain.

35. OPEN-ENDED Find values for m and b so that the

solution of the equation mx + b = − 2x − 1 is

x = −3.

36. HOW DO YOU SEE IT? The graph shows the total

revenue and expenses of a company x years after it

opens for business.

200

2

4

6

4 6 8 10

Mill

ion

s o

f d

olla

rs

x

y

Year

Revenue and Expenses

revenue

expenses

a. Estimate the point of intersection of the graphs.

b. Interpret your answer in part (a).

37. MATHEMATICAL CONNECTIONS The value of the perimeter of

the triangle (in feet) is equal

to the value of the area of

the triangle (in square feet).

Use a graph to fi nd x.

38. THOUGHT PROVOKING A car has an initial value

of $20,000 and decreases in value at a rate of

$1500 per year. Describe a different car that will be

worth the same amount as this car in exactly 5 years.

Specify the initial value and the rate at which the

value decreases.

39. ABSTRACT REASONING Use a graph to determine the

sign of the solution of the equation ax + b = cx + d

in each situation.

a. 0 < b < d and a < c b. d < b < 0 and a < c

6 ftx ft

(x − 2) ft

Section 5.6 Linear Inequalities in Two Variables 251

5.6 Linear Inequalities in Two Variables

Essential QuestionEssential Question How can you write and graph a linear inequality

in two variables?

A solution of a linear inequality in two variables is an ordered pair (x, y) that makes

the inequality true. The graph of a linear inequality in two variables shows all the

solutions of the inequality in a coordinate plane.

Writing a Linear Inequality in Two Variables


a. Write an equation represented by

the dashed line.

b. The solutions of an inequality are

represented by the shaded region. In words,

describe the solutions of the inequality.

c. Write an inequality represented by the graph.

Which inequality symbol did you use?

Explain your reasoning.

Graphing Linear Inequalities in Two Variables

Work with a partner. Graph each linear inequality in two variables. Explain your

steps. Use a graphing calculator to check your graphs.

a. y > x + 5 b. y ≤ − 1 — 2 x + 1 c. y ≥ −x − 5

Communicate Your AnswerCommunicate Your Answer 4. How can you write and graph a linear inequality in two variables?

5. Give an example of a real-life situation that can be modeled using a linear

inequality in two variables.

SELECTING TOOLSTo be profi cient in math, you need to use technological tools to explore and deepen your understanding of concepts.

x

y

2

4

2 4−2−4

−2


Work with a partner. Use a graphing calculator to graph y ≥ 1 — 4 x − 3.

a. Enter the equation y = 1 —

4 x − 3 into your calculator.

b. The inequality has the symbol ≥. So, the

region to be shaded is above the graph of

y = 1 —

4 x − 3, as shown. Verify this by testing

a point in this region, such as (0, 0), to make

sure it is a solution of the inequality.

Because the inequality symbol is greater than or equal to, the line is solid and not

dashed. Some graphing calculators always use a solid line when graphing inequalities.

In this case, you have to determine whether the line should be solid or dashed, based

on the inequality symbol used in the original inequality.

10

−10

−10

10

y ≥ x − 314

A.2.HA.3.D



5.6 Lesson What You Will LearnWhat You Will Learn Check solutions of linear inequalities.

Graph linear inequalities in two variables.

Write linear inequalities in two variables.

Use linear inequalities to solve real-life problems.

Linear InequalitiesA linear inequality in two variables, x and y, can be written as

ax + by < c ax + by ≤ c ax + by > c ax + by ≥ c

where a, b, and c are real numbers. A solution of a linear inequality in two variables

is an ordered pair (x, y) that makes the inequality true.

linear inequality in two variables, p. 252solution of a linear inequality in two variables, p. 252graph of a linear inequality, p. 252half-planes, p. 252

Previousordered pair


Checking Solutions

Tell whether the ordered pair is a solution of the inequality.

a. 2x + y < −3; (−1, 9) b. x − 3y ≥ 8; (2, −2)

SOLUTION

a. 2x + y < −3 Write the inequality.

2(−1) + 9 <?

−3 Substitute −1 for x and 9 for y.

7 < −3 ✗ Simplify. 7 is not less than −3.

So, (−1, 9) is not a solution of the inequality.

b. x − 3y ≥ 8 Write the inequality.

2 − 3(−2) ≥?

8 Substitute 2 for x and −2 for y.

8 ≥ 8 ✓ Simplify. 8 is equal to 8.

So, (2, −2) is a solution of the inequality.


Tell whether the ordered pair is a solution of the inequality.

1. x + y > 0; (−2, 2) 2. 4x − y ≥ 5; (0, 0)

3. 5x − 2y ≤ −1; (−4, −1) 4. −2x − 3y < 15; (5, −7)

Graphing Linear Inequalities in Two VariablesThe graph of a linear inequality in two variables shows all the solutions of the

inequality in a coordinate plane.

x

y4

2

2−2

All solutions of < 2lie on one side of the = 2 .

The boundary line dividesthe coordinate plane into twohalf-planes. The shadedhalf-plane is the graph of < 2 .

y

y

x

xy x

boundary line

READINGA dashed boundary line means that points on the line are not solutions. A solid boundary line means that points on the line are solutions.


Graphing a Linear Inequality in One Variable

Graph y ≤ 2 in a coordinate plane.

SOLUTION

Step 1 Graph y = 2. Use a solid line because the

x

y

1

3

2 4−1

(0, 0)

inequality symbol is ≤.

Step 2 Test (0, 0).

y ≤ 2 Write the inequality.

0 ≤ 2 ✓ Substitute.

Step 3 Because (0, 0) is a solution, shade the

half-plane that contains (0, 0).

Check

3

−1

−2

5

Core Core ConceptConceptGraphing a Linear Inequality in Two VariablesStep 1 Graph the boundary line for the inequality. Use a dashed line for < or >.

Use a solid line for ≤ or ≥.

Step 2 Test a point that is not on the boundary line to determine whether it is a

solution of the inequality.

Step 3 When the test point is a solution, shade the half-plane that contains the

point. When the test point is not a solution, shade the half-plane that

does not contain the point.

Graphing a Linear Inequality in Two Variables

Graph −x + 2y > 2 in a coordinate plane.

SOLUTION

Step 1 Graph −x + 2y = 2, or y = 1 —

2 x + 1. Use a

x

y

2

4

2−2

(0, 0)

dashed line because the inequality symbol is >.

Step 2 Test (0, 0).

−x + 2y > 2 Write the inequality.

−(0) + 2(0) >?

2 Substitute.

0 > 2 ✗ Simplify.

Step 3 Because (0, 0) is not a solution, shade the

half-plane that does not contain (0, 0).


Graph the inequality in a coordinate plane.

5. y > −1 6. x ≤ −4

7. x + y ≤ −4 8. x − 2y < 0

STUDY TIPIt is often convenient to use the origin as a test point. However, you must choose a different test point when the origin is on the boundary line.


Writing Linear Inequalities in Two Variables

Core Core ConceptConceptWriting a Linear Inequality in Two Variables Using a GraphWrite an equation in slope-intercept form of the boundary line.

If the shaded half-plane is above the boundary line, then

• replace = with > when the boundary line is dashed.

• replace = with ≥ when the boundary line is solid.

If the shaded half-plane is below the boundary line, then

• replace = with < when the boundary line is dashed.

• replace = with ≤ when the boundary line is solid.

Writing a Linear Inequality Using a Graph

Write an inequality that represents the graph.

SOLUTION

The boundary line has a slope of −2 and a y-intercept of −1. So, an equation of the

boundary line is y = −2x − 1. The shaded half-plane is below the boundary line and

the boundary line is solid. So, replace = with ≤.

The inequality y ≤ −2x − 1 represents the graph.

Writing a Linear Inequality Using a Table

An ice cream truck can carry at most 75 gallons of ice cream. The table shows the

inventory on the truck. Write an inequality that represents the numbers of gallons of

strawberry and banana ice cream on the truck.

SOLUTION

Use the table to write an inequality that represents the problem.

20 + 15 + x + y + 18 ≤ 75 The total number of gallons is less than or equal to 75.

x + y ≤ 22 Isolate the variable terms on one side.

The inequality x + y ≤ 22 represents the numbers of gallons of strawberry and

banana ice cream on the truck.


9. Write an inequality that represents the graph.

10. For a weight loss competition, the blue team needs to lose more than 55 pounds to

win. Write an inequality that represents the amounts (in pounds) of weight Team

Members B and E must lose so that the blue team wins.

Blue team member A B C D E F

Weight lost (pounds) 10 y 8 13 x 5−2

2

4

y

2 x

FlavorQuantity (gallons)

vanilla 20

chocolate 15

strawberry x

banana y

sherbet 18

Boundary Line

Dashed Solid

Hal

f-p

lan

e Above > ≥

Below < ≤

−2

−2

2

y

2 x




You can spend at most $10 on grapes and apples for a fruit salad. Grapes cost

$2.50 per pound, and apples cost $1 per pound. Write and graph an inequality that

represents the amounts of grapes and apples you can buy. Identify and interpret

two solutions of the inequality.

SOLUTION

1. Understand the Problem You know the most that you can spend and the prices

per pound for grapes and apples. You are asked to write and graph an inequality

and then identify and interpret two solutions.

2. Make a Plan Use a verbal model to write an inequality that represents the

problem. Then graph the inequality. Use the graph to identify two solutions.

Then interpret the solutions.


Words Cost per

pound of

grapes ⋅

Pounds

of grapes +

Cost per

pound of

apples⋅

Pounds

of apples ≤

Amount

you can

spend

Variables Let x be pounds of grapes and y be pounds of apples.

Inequality 2.50 ⋅ x + 1 ⋅ y ≤ 10

Step 1 Graph 2.5x + y = 10, or y = −2.5x + 10. Use a solid line because the

inequality symbol is ≤. Restrict the graph to positive values of x and y

because negative values do not make sense in this real-life context.

Step 2 Test (0, 0).

2.5x + y ≤ 10 Write the inequality.

2.5(0) + 0 ≤?

10 Substitute.

0 ≤ 10 ✓ Simplify.

Step 3 Because (0, 0) is a solution, shade the half-plane that contains (0, 0).

One possible solution is (1, 6) because it lies in the shaded half-plane. Another

possible solution is (2, 5) because it lies on the solid line. So, you can buy

1 pound of grapes and 6 pounds of apples, or 2 pounds of grapes and

5 pounds of apples.

4. Look Back Check your solutions by substituting them into the original inequality,

as shown.


11. You can spend at most $12 on red peppers and tomatoes for salsa. Red peppers

cost $4 per pound, and tomatoes cost $3 per pound. Write and graph an inequality

that represents the amounts of red peppers and tomatoes you can buy. Identify and

interpret two solutions of the inequality.

Check

2.5x + y ≤ 10

2.5(1) + 6 ≤?

10

8.5 ≤ 10 ✓

2.5x + y ≤ 10

2.5(2) + 5 ≤?

10

10 ≤ 10 ✓

Fruit Salad

2 4 60 3 51 x

1

2

3

4

5

6

7

8

9

10

0

y

(1, 6)

(2, 5)

Pou

nd

s o

f ap

ple

s

Pounds of grapes



Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3–10, tell whether the ordered pair is a solution of the inequality. (See Example 1.)

3. x + y < 7; (2, 3) 4. x − y ≤ 0; (5, 2)

5. x + 3y ≥ −2; (−9, 2) 6. 8x + y > −6; (−1, 2)

7. −6x + 4y ≤ 6; (−3, −3)

8. 3x − 5y ≥ 2; (−1, −1) 9. −x − 6y > 12; (−8, 2)

10. −4x − 8y < 15; (−6, 3)

In Exercises 11−16, tell whether the ordered pair is a solution of the inequality whose graph is shown.

11. (0, −1) 12. (−1, 3)

x

y

2

4

2−2

−2

13. (1, 4) 14. (0, 0)

15. (3, 3) 16. (2, 1)

17. MODELING WITH MATHEMATICS A carpenter has

at most $250 to spend on lumber. The inequality

8x + 12y ≤ 250 represents the numbers x of 2-by-8

boards and the numbers y of 4-by-4 boards the

carpenter can buy. Can the carpenter buy twelve

2-by-8 boards and fourteen 4-by-4 boards? Explain.

2 in. x 8 in. x 8 ft$8 each

4 in. x 4 in. x 8 ft$12 each

18. MODELING WITH MATHEMATICS The inequality

3x + 2y ≥ 93 represents the numbers x of multiple-

choice questions and the numbers y of matching

questions you can answer correctly to receive an A on

a test. You answer 20 multiple-choice questions and

18 matching questions correctly. Do you receive an A

on the test? Explain.

In Exercises 19–24, graph the inequality in a coordinate plane. (See Example 2.)

19. y ≤ 5 20. y > 6

21. x < 2 22. x ≥ −3

23. y > −7 24. x < 9

In Exercises 25−30, graph the inequality in a coordinate plane. (See Example 3.)

25. y > −2x − 4 26. y ≤ 3x − 1

27. −4x + y < −7 28. 3x − y ≥ 5

29. 5x − 2y ≤ 6 30. −x + 4y > −12

ERROR ANALYSIS In Exercises 31 and 32, describe and correct the error in graphing the inequality.

31. y < −x + 1

x

y

3

3−2

−2

✗

32. y ≤ 3x − 2

x

y4

2

2−2 −1

✗

1. VOCABULARY How can you tell whether an ordered pair is a solution of a linear inequality?

2. WRITING Compare the graph of a linear inequality in two variables with the graph of a linear

equation in two variables.



In Exercises 33–38, write an inequality that represents the graph. (See Example 4.)

33.

x

y

2

4

2−2

34.

x

y

1

4

2−2

35.

x

y1

2−1

−3

−5

36.

x

y1

2−2

−2

37.

x

y

−20

30

−20 30

38.

x

y

4

8

−4

−4 4

39. MODELING WITH MATHEMATICS A restaurant

employee works at least 40 hours per week. The table

shows the employee’s weekly work schedule. Write

an inequality that represents the numbers of hours the

employee spends cooking and washing dishes.

(See Example 5.)

Task Hours

cleaning 6

cooking x

ordering inventory 2

stocking food 3

washing dishes y

40. MODELING WITH MATHEMATICS You are allowed

to watch TV for less than 10 hours total on school

nights. The table shows the numbers of hours you

watch TV. Write an inequality that represents the

numbers of hours you can watch TV on Sunday

and Wednesday.

School night Hours

Sunday x

Monday 1.5

Tuesday 0.5

Wednesday y

Thursday 2.5

41. MODELING WITH MATHEMATICS A department store

offers a 10% discount on any purchase over $200.

The table shows the costs of items in a purchase at the

store. Write an inequality that represents the costs of

the shirt and shoes that will qualify the purchase for

the discount.

Item Cost

backpack $54.99

jeans $39.99

shirt x

shoes y

video game $49.95

ERROR ANALYSIS In Exercises 42 and 43, describe and correct the error in writing the inequality.

42.

x

y

2

5

−2 2

y > −3x + 2✗43. You eat no more than 2500 calories a day. The table

shows the numbers of calories you eat at each meal.

Meal Calories

breakfast 770

lunch x

snack 150

dinner y

snack 250

x + y ≤ ≤ 3670✗ 44. MODELING WITH MATHEMATICS You have at

most $20 to spend at an arcade. Arcade games cost

$0.75 each, and snacks cost $2.25 each. Write and

graph an inequality that represents the numbers of

games you can play and snacks you can buy. Identify

and interpret two solutions of the inequality.

(See Example 6.)


Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyWrite the next three terms of the arithmetic sequence. (Section 4.7)

55. 0, 8, 16, 24, 32, . . . 56. −5, −8, −11, −14, −17, . . .

57. 3.6, 2.8, 2, 1.2, 0.4, . . . 58. − 3 — 2 , −

1 —

2 ,

1 —

2 ,

3 —

2 ,

5 —

2 , . . .


45. MODELING WITH MATHEMATICS A drama club

must sell at least $1500 worth of tickets to cover the

expenses of producing a play. Write and graph an

inequality that represents how many adult and student

tickets the club must sell. Identify and interpret two

solutions of the inequality.

46. MODELING WITH MATHEMATICS A shipping

company is delivering ceramic pots. The company

earns $1 for each unbroken ceramic pot and is fi ned

$4 for each broken ceramic pot. The company wants

to earn at least $2000 for this delivery. Write and

graph an inequality that represents the numbers of

unbroken and broken ceramic pots the company must

deliver. Identify and interpret two solutions of the

inequality.

47. MODELING WITH MATHEMATICS A clothing store

sells T-shirts for $20 and skirts for $25. The store

wants to sell a minimum of $800 worth of T-shirts and

skirts. Write and graph an inequality that represents

the numbers of T-shirts and skirts the store must sell.

Identify and interpret two solutions of the inequality.

48. PROBLEM SOLVING Large boxes weigh 75 pounds,

and small boxes weigh 40 pounds.

a. Write and graph

an inequality that

represents the numbers

of large and small boxes

a 200-pound delivery

person can take on

the elevator.

b. Explain why some

solutions of the

inequality might not

be practical in real life.

49. WRITING Can you always use (0, 0) as a test point

when graphing an inequality? Explain.

50. HOW DO YOU SEE IT? Match each inequality with

its graph.

a. 3x − 2y ≤ 6 b. 3x − 2y < 6

c. 3x − 2y > 6 d. 3x − 2y ≥ 6

A.

x

y1

1 3−2

−2

B.

x

y1

1 3−2

−2

C.

x

y1

1 3−2

−2

D.

x

y1

1 3−2

−2

51. REASONING When graphing a linear inequality in

two variables, why must you choose a test point that

is not on the boundary line?

52. THOUGHT PROVOKING Write a linear inequality in

two variables that has the following two properties.

• (0, 0), (0, −1), and (0, 1) are not solutions.

• (1, 1), (3, −1), and (−1, 3) are solutions.

CRITICAL THINKING In Exercises 53 and 54, write and graph an inequality whose graph is described by the given information.

53. The points (2, 5) and (−3, −5) lie on the boundary

line. The points (6, 5) and (−2, −3) are solutions of

the inequality.

54. The points (−7, −16) and (1, 8) lie on the boundary

line. The points (−7, 0) and (3, 14) are not solutions

of the inequality.

Weight limit:2000 lb

Section 5.7 Systems of Linear Inequalities 259

Systems of Linear Inequalities5.7

Essential QuestionEssential Question How can you graph a system of

linear inequalities?

Graphing Linear Inequalities

Work with a partner. Match each linear inequality with its graph. Explain

your reasoning.

2x + y ≤ 4 Inequality 1

2x − y ≤ 0 Inequality 2

A.

x

y

4

2

−2

−4

1 4−2−4

B.

x

y

4

2

−4

2 4−2−4

Graphing a System of Linear Inequalities

Work with a partner. Consider the linear inequalities given in Exploration 1.

2x + y ≤ 4 Inequality 1

2x − y ≤ 0 Inequality 2

a. Use two different colors to graph the inequalities in the same coordinate plane.

What is the result?

b. Describe each of the shaded regions of the graph. What does the unshaded

region represent?

Communicate Your AnswerCommunicate Your Answer 3. How can you graph a system of linear inequalities?

4. When graphing a system of linear inequalities, which region represents the

solution of the system?

5. Do you think all systems of linear inequalities

have a solution? Explain your reasoning.

6. Write a system of linear inequalities

represented by the graph.

x

y

4

6

2

−4

−2

1 4−2−4

USING PROBLEM-SOLVING STRATEGIES

To be profi cient in math, you need to explain to yourself the meaning of a problem.

A.3.H



5.7 Lesson What You Will LearnWhat You Will Learn Check solutions of systems of linear inequalities.

Graph systems of linear inequalities.

Write systems of linear inequalities.

Use systems of linear inequalities to solve real-life problems.

Systems of Linear InequalitiesA system of linear inequalities is a set of two or more linear inequalities in the same

variables. An example is shown below.

y < x + 2 Inequality 1

y ≥ 2x − 1 Inequality 2

A solution of a system of linear inequalities in two variables is an ordered pair that

is a solution of each inequality in the system.

system of linear inequalities, p. 260solution of a system of linear inequalities, p. 260graph of a system of linear inequalities, p. 261

Previouslinear inequality in two variables


Checking Solutions

Tell whether each ordered pair is a solution of the system of linear inequalities.

y < 2x Inequality 1

y ≥ x + 1 Inequality 2

a. (3, 5) b. (−2, 0)

SOLUTION

a. Substitute 3 for x and 5 for y in each inequality.

Inequality 1 Inequality 2

y < 2x y ≥ x + 1

5 <?

2(3) 5 ≥?

3 + 1

5 < 6 ✓ 5 ≥ 4 ✓ Because the ordered pair (3, 5) is a solution of each inequality, it is a solution

of the system.

b. Substitute −2 for x and 0 for y in each inequality.

Inequality 1 Inequality 2

y < 2x y ≥ x + 1

0 <?

2(−2) 0 ≥?

−2 + 1

0 < −4 ✗ 0 ≥ −1 ✓ Because (−2, 0) is not a solution of each inequality, it is not a solution of

the system.


Tell whether the ordered pair is a solution of the system of linear inequalities.

1. (−1, 5); y < 5

y > x − 4 2. (1, 4);

y ≥ 3x + 1

y > x − 1


Graphing a System of Linear Inequalities

Graph the system of linear inequalities.

y ≤ 3 Inequality 1

y > x + 2 Inequality 2

SOLUTION

Step 1 Graph each inequality.

Step 2 Find the intersection of

the half-planes. One

solution is (−3, 1).

Core Core ConceptConceptGraphing a System of Linear InequalitiesStep 1 Graph each inequality in the same

coordinate plane.

Step 2 Find the intersection of the half-planes

that are solutions of the inequalities. This

intersection is the graph of the system.

Graphing a System of Linear Inequalities: No Solution


2x + y < −1 Inequality 1

2x + y > 3 Inequality 2

SOLUTION


Step 2 Find the intersection of the half-planes. Notice that the lines are parallel,

and the half-planes do not intersect.

So, the system has no solution.



3. y ≥ −x + 4 4. y > 2x − 3 5. −2x + y < 4

x + y ≤ 0 y ≥ 1 — 2 x + 1 2x + y > 4

Graphing Systems of Linear InequalitiesThe graph of a system of linear inequalities is the graph of all the solutions of

the system.

x

y

4

6

2 4−1

y < x + 2y ≥ 2x − 1

x

y

4

1

−2

2−1−4

(−3, 1)

The solution is thepurple-shaded region.

Check

Verify that (−3, 1) is a

solution of each inequality.

Inequality 1

y ≤ 3

1 ≤ 3 ✓Inequality 2

y > x + 2

1 >?

−3 + 2

1 > −1 ✓

x

y

2

−2

−4

1 3−2−4


Writing Systems of Linear Inequalities

Writing a System of Linear Inequalities

Write a system of linear inequalities represented

by the graph.

SOLUTION

Inequality 1 The horizontal boundary line passes

through (0, −2). So, an equation of the

line is y = −2. The shaded region is

above the solid boundary line, so the

inequality is y ≥ −2.

Inequality 2 The slope of the other boundary line is 1, and the y-intercept

is 0. So, an equation of the line is y = x. The shaded region

is below the dashed boundary line, so the inequality is y < x.

The system of linear inequalities represented by the graph is

y ≥ − 2 Inequality 1

y < x. Inequality 2

x

y

2

−4

2 4−2−4

Writing a System of Linear Inequalities

Write a system of linear inequalities represented

by the graph.

SOLUTION

Inequality 1 The vertical boundary line passes

through (3, 0). So, an equation of the

line is x = 3. The shaded region is to

the left of the solid boundary line,

so the inequality is x ≤ 3.

Inequality 2 The slope of the other boundary line is 2 —

3 , and the y-intercept is −1.

So, an equation of the line is y = 2 —

3 x − 1. The shaded region is above

the dashed boundary line, so the inequality is y > 2 —

3 x − 1.

The system of linear inequalities represented by the graph is

x ≤ 3 Inequality 1

y > 2 —

3 x − 1. Inequality 2


Write a system of linear inequalities represented by the graph.

6.

x

y

1

2 4

−2

7.

x

y

2

4

2 4

−2

x

y

2

4

−4

2 4 6




You have at most 8 hours to spend at the mall

and at the beach. You want to spend at least

2 hours at the mall and more than 4 hours

at the beach. Write and graph a system that

represents the situation. How much time can

you spend at each location?

SOLUTION

1. Understand the Problem You know the total amount of time you can spend

at the mall and at the beach. You also know how much time you want to spend

at each location. You are asked to write and graph a system that represents the

situation and determine how much time you can spend at each location.

2. Make a Plan Use the given information to write a system of linear inequalities.

Then graph the system and identify an ordered pair in the solution region.

3. Solve the Problem Let x be the number of hours at the mall and let y be the

number of hours at the beach.

x + y ≤ 8 at most 8 hours at the mall and at the beach

x ≥ 2 at least 2 hours at the mall

y > 4 more than 4 hours at the beach

Graph the system.

Time at the Mall and at the Beach

2 4 6 7 8 90 3 51 x

1

2

3

4

5

6

7

8

0

y

Ho

urs

at

the

bea

ch

Hours at the mall

One ordered pair in the solution region is (2.5, 5).

So, you can spend 2.5 hours at the mall and 5 hours at the beach.

4. Look Back Check your solution by substituting it into the inequalities in the

system, as shown.


8. Name another solution of Example 6.

9. WHAT IF? You want to spend at least 3 hours at the mall. How does this change

the system? Is (2.5, 5) still a solution? Explain.

Check

x + y ≤ 8

2.5 + 5 ≤?

8

7.5 ≤ 8 ✓

x ≥ 2

2.5 ≥ 2 ✓

y > 4

5 > 4 ✓




Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with MathematicsIn Exercises 3−6, tell whether the ordered pair is a solution of the system of linear inequalities.

3. (−4, 3)

4. (−3, −1)

5. (−2, 5)

6. (1, 1)

In Exercises 7−10, tell whether the ordered pair is a solution of the system of linear inequalities. (See Example 1.)

7. (−5, 2); y < 4

y > x + 3 8. (1, −1);

y > −2

y > x − 5

9. (0, 0); y ≤ x + 7

y ≥ 2x + 3 10. (4, −3);

y ≤ −x + 1

y ≤ 5x − 2

In Exercises 11−20, graph the system of linear inequalities. (See Examples 2 and 3.)

11. y > −3 12. y < −1

y ≥ 5x x > 4

13. y < −2 14. y < x − 1

y > 2 y ≥ x + 1

15. y ≥ −5 16. x + y > 4

y − 1 < 3x y ≥ 3 — 2 x − 9

17. x + y > 1 18. 2x + y ≤ 5

−x − y < −3 y + 2 ≥ −2x

19. x < 4 20. x + y ≤ 10

y > 1 x − y ≥ 2

y ≥ −x + 1 y > 2

In Exercises 21−26, write a system of linear inequalities represented by the graph. (See Examples 4 and 5.)

21.

x

y

2

4

2−2

22.

x

y

2

1 53−1

−2

23.

x

y

2

4−2 −1

24.

x

y

1

2−2

25.

x

y

2

−3

2−2−4

26.

x

y

2

−3

2−3

1. VOCABULARY How can you verify that an ordered pair is a solution

of a system of linear inequalities?

2. WHICH ONE DOESN’T BELONG? Use the graph shown. Which of

the ordered pairs does not belong with the other three? Explain

your reasoning.

(1, −2) (0, −4) (−1, −6) (2, −4)

x

y

−2

−4

−6

52

x

y

1−3−5

2

4


ERROR ANALYSIS In Exercises 27 and 28, describe and correct the error in graphing the system of linear inequalities.

27. y ≤ ≤ x − 1

x

y

1

−3

2−2

y ≥ ≥ x + 3✗

28. y ≤ ≤ 3x + 4

x

y

1

4

2−2−4

y > 1 — 2

x + 2✗

29. MODELING WITH MATHEMATICS You can spend at

most $21 on fruit. Blueberries cost $4 per pound, and

strawberries cost $3 per pound. You need at least

3 pounds of fruit to make muffi ns. (See Example 6.)

a. Write and graph a system of linear inequalities

that represents the situation.

b. Identify and interpret a solution of the system.

c. Use the graph to

determine whether

you can buy

4 pounds of

blueberries

and 1 pound

of strawberries.

30. MODELING WITH MATHEMATICS You earn

$10 per hour working as a manager at a grocery store.

You are required to work at the grocery store at least

8 hours per week. You also teach music lessons for

$15 per hour. You need to earn at least $120 per week,

but you do not want to work more than 20 hours

per week.



b. Identify and interpret a solution of the system.

c. Use the graph to determine whether you can work

8 hours at the grocery store and teach 1 hour of

music lessons.

31. MODELING WITH MATHEMATICS You are fi shing

for surfperch and rockfi sh, which are species of

bottomfi sh. Gaming laws allow you to catch no more

than 15 surfperch per day, no more than 10 rockfi sh

per day, and no more than 20 total bottomfi sh per day.



b. Use the graph to determine whether you can catch

11 surfperch and 9 rockfi sh in 1 day.

surfperch rockfish

32. REASONING Describe the intersection of the

half-planes of the system shown.

x − y ≤ 4

x − y ≥ 4

33. MATHEMATICAL CONNECTIONS The following points

are the vertices of a shaded rectangle.

(−1, 1), (6, 1), (6, −3), (−1, −3)

a. Write a system of linear inequalities represented

by the shaded rectangle.

b. Find the area of the rectangle.

34. MATHEMATICAL CONNECTIONS The following points

are the vertices of a shaded triangle.

(2, 5), (6, −3), (−2, −3)

a. Write a system of linear inequalities represented

by the shaded triangle.

b. Find the area of the triangle.

35. PROBLEM SOLVING You plan to spend less than

half of your monthly $2000 paycheck on housing

and savings. You want to spend at least 10% of

your paycheck on savings and at most 30% of it on

housing. How much money can you spend on savings

and housing?

36. PROBLEM SOLVING On a road trip with a friend, you

drive about 70 miles per hour, and your friend drives

about 60 miles per hour. The plan is to drive less than

15 hours and at least 600 miles each day. Your friend

will drive more hours than you. How many hours can

you and your friend each drive in 1 day?


Maintaining Mathematical ProficiencyMaintaining Mathematical ProficiencyWrite the product using exponents. (Skills Review Handbook)

49. 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 50. (−13) ⋅ (−13) ⋅ (−13) 51. x ⋅ x ⋅ x ⋅ x ⋅ x ⋅ x

Write an equation of the line with the given slope and y-intercept. (Section 4.1)

52. slope: 1 53. slope: −3 54. slope: − 1 — 4 55. slope:

4 —

3

y-intercept: −6 y-intercept: 5 y-intercept: −1 y-intercept: 0


37. WRITING How are solving systems of linear

inequalities and solving systems of linear equations

similar? How are they different?

38. HOW DO YOU SEE IT? The graphs of two linear

equations are shown.

x

y

2

−2

−4

1 3−2−4

y = −3x + 4

A B

C

Dy = 2x + 1

Replace the equal signs with inequality symbols to

create a system of linear inequalities that has point C

as a solution, but not points A, B, and D. Explain

your reasoning.

y −3x + 4

y 2x + 1

39. OPEN-ENDED Write a real-life problem that can be

represented by a system of linear inequalities. Write

the system of linear inequalities and graph the system.

40. MAKING AN ARGUMENT Your friend says that a

system of linear inequalities in which the boundary

lines are parallel must have no solution. Is your friend

correct? Explain.

41. CRITICAL THINKING Is it possible for the solution

set of a system of linear inequalities to be all real

numbers? Explain your reasoning.

OPEN-ENDED In Exercises 42−44, write a system of linear inequalities with the given characteristic.

42. All solutions are in Quadrant I.

43. All solutions have one positive coordinate and one

negative coordinate.

44. There are no solutions.

45. OPEN-ENDED One inequality in a system is

−4x + 2y > 6. Write another inequality so the system

has (a) no solution and (b) infi nitely many solutions.

46. THOUGHT PROVOKING You receive a gift certifi cate

for a clothing store and plan to use it to buy T-shirts

and sweatshirts. Describe a situation in which you can

buy 9 T-shirts and 1 sweatshirt, but you cannot buy

3 T-shirts and 8 sweatshirts. Write and graph a system

of linear inequalities that represents the situation.

47. CRITICAL THINKING Write a system of linear

inequalities that has exactly one solution.

48. MODELING WITH MATHEMATICS You make

necklaces and key chains to sell at a craft fair. The

table shows the amounts of time and money it takes to

make a necklace and a key chain, and the amounts of

time and money you have available for making them.

Necklace Key chain Available

Time to make (hours)

0.5 0.25 20

Cost to make (dollars)

2 3 120

a. Write and graph a system of four linear

inequalities that represents the number x of

necklaces and the number y of key chains that

you can make.

b. Find the vertices (corner points) of the graph of

the system.

c. You sell each necklace for $10 and each key chain

for $8. The revenue R is given by the equation

R = 10x + 8y. Find the revenue corresponding to

each ordered pair in part (b). Which vertex results

in the maximum revenue?

267

5.5–5.7 What Did You Learn?

Core VocabularyCore Vocabularylinear inequality in two variables, p. 252solution of a linear inequality in two variables,

p. 252graph of a linear inequality, p. 252

half-planes, p. 252system of linear inequalities, p. 260solution of a system of linear inequalities, p. 260graph of a system of linear inequalities, p. 261

Core ConceptsCore ConceptsSection 5.5Solving Linear Equations by Graphing, p. 246

Section 5.6Graphing a Linear Inequality in Two Variables, p. 253Writing a Linear Inequality in Two Variables, p. 254

Section 5.7Graphing a System of Linear Inequalities, p. 261Writing a System of Linear Inequalities, p. 262

Mathematical ThinkingMathematical Thinking1. Describe how to solve Exercise 39 on page 250 algebraically.

2. Why is it important to be precise when answering part (a) of Exercise 48 on page 258?

3. Describe the overall step-by-step process you used to solve Exercise 35 on page 265.

Performance Task

Prize PatrolYou have been selected to drive a prize patrol cart and place prizes on the competing teams’ predetermined paths. You know the teams’ routes and you can only make one pass. Where will you place the prizes so that each team will have a chance to fi nd a prize on their route?

To explore the answer to this question and more, go to BigIdeasMath.com.

267

Tasskk

prizeseseoua


55 Chapter Review

Solving Systems of Linear Equations by Graphing (pp. 219–224)5.1

Solve the system by graphing. y = x − 2 Equation 1


Step 1 Graph each equation.


The graphs appear to intersect at (1, −1).



y = x − 2 y = −3x + 2

−1 =?

1 − 2 −1 =?

−3(1) + 2

−1 = −1 ✓ −1 = −1 ✓ The solution is (1, −1).


1. y = −3x + 1 2. y = −4x + 3 3. 5x + 5y = 15

y = x − 7 4x − 2y = 6 2x − 2y = 10

Solving Systems of Linear Equations by Substitution (pp. 225–230)5.2

Solve the system by substitution. −2x + y = −8 Equation 1


Step 1 Solve for y in Equation 1.

y = 2x − 8 Revised Equation 1

Step 2 Substitute 2x − 8 for y in Equation 2 and solve for x.


7x + (2x − 8) = 10 Substitute 2x − 8 for y.

9x − 8 = 10 Combine like terms.

9x = 18 Add 8 to each side.

x = 2 Divide each side by 9.

Step 3 Substituting 2 for x in Equation 1 and solving for y gives y = −4.

The solution is (2, −4).


4. 3x + y = −9 5. x + 4y = 6 6. 2x + 3y = 4

y = 5x + 7 x − y = 1 y + 3x = 6

7. You spend $20 total on tubes of paint and disposable brushes for an art project. Tubes of paint

cost $4.00 each and paintbrushes cost $0.50 each. You purchase twice as many brushes as tubes

of paint. How many brushes and tubes of paint do you purchase?

x

y

2

42

(1, −1)

y = −3x + 2

y y = x − 2

−1

Chapter 5 Chapter Review 269

Solving Systems of Linear Equations by Elimination (pp. 231–236)5.3

Solve the system by elimination. 4x + 6y = −8 Equation 1 x − 2y = −2 Equation 2

Step 1 Multiply Equation 2 by 3 so that the coeffi cients of the y-terms are opposites.

4x + 6y = −8 4x + 6y = −8 Equation 1

x − 2y = −2 3x − 6y = −6 Revised Equation 2


4x + 6y = −8 Equation 1


7x = −14 Add the equations.

Step 3 Solve for x.

7x = −14 Resulting equation from Step 2


Step 4 Substitute −2 for x in one of the original equations

and solve for y.


4(−2) + 6y = −8 Substitute −2 for x.

−8 + 6y = −8 Multiply.

y = 0 Solve for y.

The solution is (−2, 0).

Solve the system of linear equations by elimination. Check your solution.

8. 9x − 2y = 34 9. x + 6y = 28 10. 8x − 7y = −3

5x + 2y = −6 2x − 3y = −19 6x − 5y = −1

Solving Special Systems of Linear Equations (pp. 237–242)5.4

Solve the system. 4x + 2y = −14 Equation 1


Solve by substitution. Substitute −2x − 6 for y in Equation 1.


4x + 2(−2x − 6) = −14 Substitute −2x − 6 for y.

4x − 4x − 12 = −14 Distributive Property

−12 = −14 ✗ Combine like terms.

The equation −12 = −14 is never true. So, the system has no solution.


11. x = y + 2 12. 3x − 6y = −9 13. −4x + 4y = 32

−3x + 3y = 6 −5x + 10y = 10 3x + 24 = 3y

Check

Equation 1

4x + 6y = −8

4(−2) + 6(0) =?

−8

−8 = −8 ✓Equation 2

x − 2y = −2

(−2) − 2(0) =?

−2

−2 = −2 ✓

Multiply by 3.


Solving Equations by Graphing (pp. 245–250)5.5

Solve 3x − 1 = −2x + 4 by graphing. Check your solution.


3x − 1 = −2x + 4−


x

y

2

4

1 3 5

y = 3x − 1

y = −2x + 4

−1

(1, 2)



The graphs intersect at (1, 2).

So, the solution of the equation is x = 1.

Solve the equation by graphing. Check your solution(s).

14. x + 1 = −x − 9 15. 2x − 8 = x + 5 16. 1 — 3 x + 5 = −2x − 2

Check

3x − 1 = −2x + 4

3(1) − 1 =?

−2(1) + 4

2 = 2 ✓

y = −2x + 4

h th t

y = 3x − 1

Linear Inequalities in Two Variables (pp. 251–258)5.6

Graph 4x + 2y ≥ ≥ −6 in a coordinate plane.

Step 1 Graph 4x + 2y = −6, or y = −2x − 3. Use a solid line

x

y1

−3

2−2

because the inequality symbol is ≥.

Step 2 Test (0, 0).

4x + 2y ≥ −6 Write the inequality.

4(0) + 2(0) ≥?

−6 Substitute.

0 ≥ −6 ✓ Simplify.

Step 3 Because (0, 0) is a solution, shade the half-plane that contains (0, 0).

Graph the inequality in a coordinate plane.

17. y > −4 18. −9x + 3y ≥ 3 19. 5x + 10y < 40

Systems of Linear Inequalities (pp. 259–266)5.7

Graph the system. y < x − 2 Inequality 1

y ≥ 2x − 4 Inequality 2


Step 2 Find the intersection of the

half-planes. One solution is (0, −3).


20. y ≤ x − 3 21. y > −2x + 3 22. x + 3y > 6

y ≥ x + 1 y ≥ 1 — 4 x − 1 2x + y < 7

x

y1

−4

3−2

(0, −3)The solution is thepurple-shaded region.

Chapter 5 Chapter Test 271

Chapter Test55Solve the system of linear equations using any method. Explain why you chose the method.

1. 8x + 3y = −9 2. 1 —

2 x + y = −6 3. y = 4x + 4

−8x + y = 29 y = 3 —

5 x + 5 −8x + 2y = 8

4. x = y − 11 5. 6x − 4y = 9 6. y = 5x − 7

x − 3y = 1 9x − 6y = 15 −4x + y = −1

7. Write a system of linear inequalities so the points (1, 2) and (4, −3) are solutions of the

system, but the point (−2, 8) is not a solution of the system.


8. y > −1 9. y > 1 —

2 x + 4

x < 3 2y ≤ x + 4

10. x + y < 1 11. y ≥ − 2 — 3 x + 1

5x + y > 4 −3x + y > −2

12. You pay $45.50 for 10 gallons of gasoline and 2 quarts of oil at a gas station.

Your friend pays $22.75 for 5 gallons of the same gasoline and 1 quart of the

same oil.

a. Is there enough information to determine the cost of 1 gallon of gasoline

and 1 quart of oil? Explain.

b. The receipt shown is for buying the same gasoline and same oil. Is there

now enough information to determine the cost of 1 gallon of gasoline and

1 quart of oil? Explain.

c. Determine the cost of 1 gallon of gasoline and 1 quart of oil.

13. Describe the advantages and disadvantages of solving a system of linear

equations by graphing.

14. You have at most $60 to spend on trophies and medals to give as

prizes for a contest.

a. Write and graph an inequality that represents the numbers of

trophies and medals you can buy. Identify and interpret a solution

of the inequality.

b. You want to purchase at least 6 items. Write and graph a system

that represents the situation. How many of each item can you buy?

15. Compare the slopes and y-intercepts of the graphs of the equations

in the linear system 8x + 4y = 12 and 3y = −6x − 15 to determine

whether the system has one solution, no solution, or infi nitely many

solutions. Explain.

Trophies$12 each

Medals$3 each


55 Standards Assessment

1. The graph of which equation is shown? (TEKS A.2.B)

x

y2

−4

−8

84

(2, 0)

(0, −9)

○A 9x − 2y = −18

○B −9x − 2y = 18

○C 9x + 2y = 18

○D −9x + 2y = −18

2. A van rental company rents out 6-, 8-, 12-, and 16-passenger vans. The function

C(x) = 100 + 5x represents the cost C (in dollars) of renting an x-passenger van for a

day. Which of the following numbers is in the range of the function? (TEKS A.2.A)

○F 6 ○G 100

○H 130 ○J 150

3. GRIDDED ANSWER The students in the graduating classes at three high schools in a

school district have to pay for their cap-and-gown sets and extra tassels. At one high

school, students pay $3262 for 215 cap-and-gown sets and 72 extra tassels. At another

high school, students pay $3346 for 221 cap-and-gown sets and 72 extra tassels. How

much (in dollars) do students at the third high school pay for 218 cap-and-gown sets

and 56 extra tassels? (TEKS A.2.I, TEKS A.5.C)

4. Which of the following points is not a solution of the system of linear inequalities

represented by the graph? (TEKS A.3.H)

○A (4, 10)

x

y

4

2

42

○B (5, 8)

○C (−2, 8)

○D (0, 10)

5. Consider the function f(x) = 2x − 1. Which of the

x

y6

2

2−4

gfollowing functions are represented by the graph? (TEKS A.3.E)

I. g(x) = f(x) + 6

II. g(x) = f(x − 3)

III. g(x) = f(x + 3)

IV. g(x) = f(2x)

○F I and II only ○G I and III only

○H I, II, and III only ○J I, II, III, and IV

Chapter 5 Standards Assessment 273

6. Line b is perpendicular to line a and passes through the point (−9, 2). Which of the following is an equation for line b? (TEKS A.2.F)

○A y = 1 — 3 x + 5

x

y

4

42−2

a

○B y = − 1 — 3 x − 1

○C y = 1 — 3 x − 29

— 3

○D y = − 1 — 3 x + 1

7. Which system of linear equations has no solution? (TEKS A.5.C)

○F y − 2x = 5 ○G y = −4x + 6 4y − 10x = 20 y = 8x − 12

○H 2y = 6x + 4 ○J y + 1 = 4x y − 1 —

2 = 3x y + 1 = 1 —

4 x

8. A car dealership offers interest-free car loans for one day only. During this day, a

salesperson at the dealership sells two cars. One of the clients decides to pay off

a $17,424 car in 36 monthly payments of $484. The other client decides to pay off a

$15,840 car in 48 monthly payments of $330. Which system of equations can you use

to determine the number x of months after which both clients will have the same loan

balance y (in dollars)? (TEKS A.2.I)

○A y = −484x ○B y = −484x + 17,424 y = −330x y = −330x + 15,840

○C y = −484x + 15,840 ○D y = 484x + 17,424 y = −330x + 17,424 y = 330x + 15,840

9. The graph of which linear inequality is shown? (TEKS A.2.H)

○F y < −2x − 1

x

y

2

−2

2−2

○G y ≤ −2x − 1

○H y > −2x − 1

○J y ≥ −2x − 1

10. Simplify the expression 4(2x + 4) − 6(x + 1). (TEKS A.10.D)

○A 2x + 10 ○B 2x + 15

○C 2x + 17 ○D 2x + 22

5Solving Systems of Linear Equations - Big Ideas …...5Solving Systems of Linear Equations 5.1 Solving Systems of Linear Equations by Graphing 5.2 Solving Systems of Linear Equations

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